No, they do not. As far as ancient sources go, none of these is explicitly mentioned as having such a birth-date.
The earliest source I can find making this claim in reference to Mithras goes back to 1795, in L'origine de tous les cultes, ou la réligion universelle, wherein Charles-François Dupuis also interprets Chapter 22 of Firmicus Maternus's book On the Error of the Profane Religions in such a manner as to give us the only ancient allusion to a death and resurrection of this god. (As we shall note later, though, Maternus does not name Mithras, or any other deity, for that matter, in said chapter. See below, under the section Origins: Coming Up with a Death Scene for Mithras.)
The birthday information is acquired by conflating Mithras with the official Roman god of the sun, in which methodology certain scholars would come to follow Dupuis. Notable among them is Franz Cumont, since he is credited with having founded modern Mithraic studies, although at present day (about a hundred years following the initial release of Cumont's works) certain other academics have their arguments against the connection of the date with Mithras.
As for Dionysus, I think that the forensic thread also leads back to Dupuis's work L'origine, and the rather particular interpretation he makes therein of a passage from Book 1 of Macrobius' Saturnalia. Starting at least around the middle of the 1800s, literature focused on comparing Christianity with other religions and their sacred stories would become a veritable genre produced in Western Europe and the US. All of the texts of this nature that I'm finding, if they don't cite Dupuis directly, then refer to someone else who in turn points back to the said 18th-century French publication.
One such effort is entitled Anacalypsis (1836), by an Englishman named Godfrey Higgins,1 and is the oldest reference I can find that outright says that December 25th is the birthday of Dionysus, or rather of his Roman counterpart Bacchus. Incidentally this text this is also the oldest place in which I'm able to find the same claim about Osiris and Adonis. The foundation for the assertion regarding Adonis is a lot shiftier than the one about Dionysus. Nonetheless, by looking at Higgins' purported sources, I believe I can see the seams where he stitches some ideas together to arrive at this conclusion.
When it comes to Osiris, meanwhile, we're much closer to the realm of just making stuff up, and the author even skates quite close to admitting as much. There is even less reliability when it comes to Krishna, whom he says (on p. 98 of Vol. 2) was "born in the eighth month, which answers to our December". This is incorrect, as this divinity's birthday is actually celebrated in a festival in Bhādra, the 6th month of the Hindu calendar, on a day landing in, depending on the year, either late August or early September of the Gregorian calendar. But matters only get worse with subsequent writers on the subject, who cite Anacalypsis unabashedly.
Dionysus
In his Saturnalia, Macrobius argues (in Chapter 18 of Book 1) that Apollo and Bacchus/Dionysus are different aspects of the same sun-god. According to §10 of the chapter, Dionysus is represented in statues
sometimes as a child, and sometimes as a young man; again, as a
man with a beard and also as an old man... These differences in age
have reference to the sun, for at the winter solstice the sun would
seem to be a little child, like that which the Egyptians bring forth
from a shrine on an appointed day, since the day is then at its
shortest and the god is accordingly shown as a tiny infant.2
Macrobius fails to specify which "appointed day" is meant here, and neglects to inform us which Egyptian deity he has in view. However, because the divinity is mentioned in connection with Bacchus-Dionysus, the interpretation of Dupuis, in L'origine, is that an image specifically of Bacchus in the form of a child "had been exposed to the eyes of the Initiates at the time of 25 December".3
Fast-forward a few decades, and when Higgins paraphrases this passage from L'origine in his Anacalypsis (Vol. 2, p. 102), he cites Dupuis and then just plainly states that Bacchus "was born of a virgin on the 25th of December". As we shall see below, around the time of Macrobius the winter solstice in the Late Antique Roman Empire was actually calculated to be a few days before the 25th (see the reference to the parapegma of Antiochus of Athens under the section Sol Invictus, and Mithras: The Sources for the Birthday of Invictus). But between Dupuis and Higgins (the latter especially) such nuances (like the fact that no version of Bacchus was ever "born of a virgin") are evidently inconsequential.
Osiris
Unlike with pretty much all the other figures in your title, the actual date of Osiris's birth itself is a big deal in ancient myth and ritual, because when his mother was pregnant with him and his four siblings, the Sun had resentfully forbidden her to give birth on any day of the year, which at that point was composed of only 360 days. By winning a gambling contest against the Moon, Hermes (usually interpreted to be the Egyptian god Thoth) was able to squeeze five extra days into the calendar by earning the moon-light necessary to create them. These five days were then used to recalibrate the Egyptian calendar every year.
The story is told by Plutarch in On Isis and Osiris, found in his collection of essays entitled [in Greek] Ethika (or Moralia as it is called in Latin), in which he says that Osiris was born on the first of these additional days. In the Coptic calendar used in Egypt at present day, and descended from the Ancient Egyptian one, this first day corresponds to September 6th in the modern Gregorian calendar (= August 24th in the Julian calendar). Just as in the Coptic, in the Ethiopian calendar, it makes up the beginning of the 13th "add-on" month of the year, which is only 5 days long, and which would have occurred around June or July in the ancient version of these dates.
In the same narrative about the birth of Osiris and his siblings, Plutarch says that the Greeks came to identify this god with Dionysus. If we insisted that therefore we know Dionysus' birthday based on it being the same as that of Osiris, i.e. the 1st of the 5 intercalary days, this still gives us, as above, sometime in June/July.
It's probably by splicing together Dupuis's reading of Saturnalia 1.18.10 (as above, that Bacchus was born at the winter solstice), with this apparent identification of Osiris and Dionysus, that one could arrive at anything resembling direct evidence of a December date for Osiris's birth.
Sol Invictus, and Mithras
Mithras was identified with the sun and regularly labelled in votive inscriptions as Deus Sol Invictus Mithras, "the God, Unconquered Sun, Mithras." Frequently, however, Mithras is also clearly represented as a separate, distinct entity from the sun, depicted in the company of the sun-god Sol Invictus, such as when the two of them banquet together on the bull slaughtered by Mithras, or when Sol gives Mithras a lift in his chariot.
These scenes form some of the typical imagery featured at the sites known as Mithraea, from their use in the mysteries of Mithras, an esoteric cult restricted to small groups of men (i.e. adult males exclusively) spread out all over the Roman Empire. Owing to the secretive nature of this cult, just as in other mysteries in the empire, and in ancient times generally, as well as throughout the world across history, little is currently known with certainty regarding the precise meaning of these images, or regarding the activities in which the cult’s initiates engaged themselves. Conversely, the public cult of Sol Invictus, practised out in the open, just as those of the other gods in general, included every inhabitant of the empire by default.
The Sources for the Birthday of Invictus
Technically there is no ancient source that explicitly mentions a birth-date for Mithras specifically referring to him by name.
- In the part of the Chronography of AD 354 known as the Philocalian
Calendar, December 25th is marked as the birthday of Invictus, the
“Invincible” One, without attaching an actual name to the epithet.
According to the annotation, thirty races were to be conducted in the
Circus on that day.
- In an anonymous Christian homily, supposedly from the same century as
the 354 Chronograph, and entitled On the Solstices and the
Equinoxes of the Conception and Nativity of our Lord Jesus Christ and
of John the Baptist, it is said that “they call it [December 25th]
the Birthday of Invictus” and also “the Birthday of Sol [or of the
Sun]”. (In the time of Dupuis it was thought that St John Chrysostom was the author of this text, an attribution now understood to be spurious.)
- An Egyptian calendar of uncertain date, written in Greek, and known as
the parapegma of Antiochus of Athens, lists astronomical events, such
as “Winter Solstice” for December 22nd, and “[the star] Prokyon
[Procyon] sets in the east” for December 23rd. For December 25th it
says: “Birth of Helios [or of the Sun]; light increases”.
- According to Emperor Julian the Apostate’s Oration 4, also called
the Hymn to King Helios (composed less than a decade after the
Philocalian Calendar), there was, sometime between December 23rd and
the beginning of the New Year, a festival dedicated to Helios
Aniketos [Anicetus], which is the Greek form of the name of Sol
Invictus.
In his 2011 essay “Usener’s Christmas: A Contribution to the Modern Construct of Late Antique Solar Syncretism,” Steven Hijmans says that this documentation of the Sun’s birthday
is often expanded with the additional claims… that in the cult of
Mithras there was also a major festival on that day. The ease with
which [this is] often repeated is surprising. Of the mystery cult of
Sol Invictus Mithras we know little with certainty, and even if we
leave aside the problem of the relationship between the Mithraic
mysteries and the public cult of Sol, the notion that Mithraists
celebrated December 25 in some fashion is a modern invention for which
there is simply no evidence.
Roger Beck critiques the notion that Mithras was born on December 25 by making a distinction between, on the one hand, the smaller-scale, secretive Mithraic cult and, on the other hand, the Natalis Invicti ("Birthday of Invictus") festival attested in the 354 Chronograph, celebrated on a large scale with public games such as it was, and thus part of the sanctioned business of the empire regarding the state’s official sun-deity, the Titan Sol. Based on this contrast, with Mithras being an unofficial, privately-venerated god, Beck surmises that “it does not follow that” Mithras himself was also “necessarily or even probably, born on” the same day as the Sun-Titan.4
An analogy to expand on the point being made here, as I understand it, might be a look at yet another god who was syncretised with Sol/Helios, and whose birthday is likewise anciently attested, namely Apollo. According to Plutarch, this deity was called Hebdomagenes, from having been born on the seventh day of the month. (At Delos this was the month of Thargelion while at Delphi the month was Bysios.) It would likely be misleading and inaccurate to make the assumption that Sol, because he was so closely linked with Apollo as a fellow god of light, was therefore also thought to have been born on the seventh day of the month.
Moreover, since Sol and Mithras were often identical in the context of the exclusivist Mithraic mysteries, why not go ahead and assume that Mithras too was born on this seventh day? (According to Pseudo-Clementine Homily 7.10, Apollo was “a son of Zeus, who was also called Mithras”.) This would be just valid (or rather invalid?) as saying that both Apollo and Mithras surely were born on the date in question in December since they and Sol Invictus are all, in some way, the sun.
ORIGINS
Coming Up with a Death Scene for Mithras
In 1872 the abridged version of Dupuis' aforementioned work was translated into English as The Origin of All Religious Worship. The claim that Mithras was born on December 25 appears on p. 246 of the publication, in the same passage asserting that Mithras “died as he [Christ] did; he had his sepulchre, over which his disciples came to shed tears. During the night the priests carried his image to a tomb, expressly prepared for him; he was laid out on a litter, like the Phoenician Adonis.” These priests are then described as mourning for the deceased god and thereafter proclaiming rejoicing that he had risen from the dead.
This scenario is a paraphrase of the second half of On the Error of the Profane Religions 22.1, written by Firmicus Maternus, and which reads thus:
On a certain night a statue is laid flat on its back on a bier,
where it is bemoaned in cadenced plaints. Then when the worshippers
have had their fill of feigned lamentation, a light is brought in.
Next a priest anoints the throats of all who were mourning, and once
that is done he whispers in a low murmur:
“Rejoice, o initiates! Behold, our god appears as saved!
And we shall find salvation, springing from our woes.”
The passage is ambiguous, as Maternus never names the deity to whom he is referring in this section, which leaves it open to interpretation. Maternus's 1970 translator Clarence A. Forbes relates some mild scholarly controversy over the matter, naming five academics (one of them being Franz Cumont) who favour Osiris for the figure's identity, "Though Heuten and several others have argued for Attis," while Adonis is also apparently considered as an alternate candidate.5
Referring to another ritual narrated further down in On the Error 22, Nilsson, one of the aforementioned five, "observed that to have a god's stone image dismembered and then put together again (22.3) was a cult action appropriate to Osiris and not to Attis."6 Dupuis, a century or so earlier, is special in having authoritatively applied this passage to Mithras, founding the basis for numerous subsequent declarations that Mithras was thought to have died and resurrected.
By 1882 it would become possible to read in Bible Myths and Their Parallels in Other Religions, a book by an American named Thomas William Doane, that "Mithras remained in the grave a period of three days" [italics in original]. The only proof offered for this is a quote from another author saying: "The Persians believed that the soul of man remained yet three days in the world after its separation from the body {emphasis in original}."
For the rest of the claim he cites p. 99 of Higgins's Anacalypsis Vol. 2, which in turn says that:
The same God was believed, by the inhabitants of Persia, Asia
Minor, and Armenia, under the name of Mithra, to have been born in a
cave on the 25th of December, to have been put to death, and to have
risen again on the 25th of March."
At this point Higgins provides a full quote of Dupuis's paraphrase of the ritual narrated by Maternus (who, as noted, never says that this has to do with Mithras).
Adding Adonis & Osiris to the Concoction
As for the other divinities in question, according to Vol. 2, pp. 98-99 of Anacalypsis:
Tertullian, Jerom [sic], and other fathers of the church, inform us,
that the Gentiles celebrated, on the 25th of December… the birth of
the God Sol, under the name of Adonis, in a cave, like that of Mithra
(in Persia Mithra; in Egypt, Phœnicia, and Biblis, Adonis), and
that the cave wherein they celebrated his mysteries was that in which
Christ was born in the city of Bethlehem, or, according to the strict
meaning of the word Bethlehem, in the city of the house of the sun.
[Emphases in original]
From a footnote, Dupuis is again supposedly the one to blame for this shopping list of ideas. The birthday reference is probably alluding to the aforementioned homily On the Solstices and the Equinoxes. St Jerome does imply, in his Letter 58 (to Paulinus), that Christ was born in a cave in Bethlehem. In order to spite Christians, he says, starting in the reign of Hadrian, around this cave a grove was dedicated to lamentation for the death of Tammuz, whom he identifies with Adonis. Beyond that, as far as I know, none of the Church Fathers ever equates Adonis with either the traditional Greco-Roman sun-god or with any version of Mithras, whether in his Persian or Romanised guise.
Further down p. 99 of Anacalypsis 2:
At the first moment after midnight of the 24th of December, all
the nations of the earth, by common consent, celebrated the
accouchement of the Queen of Heaven, of the Celestial Virgin of the
sphere, and the birth of the God Sol, the infant Orus or Aur, the God
of Day...
The Egyptians celebrated the birth of the son of Isis on the 25th
of December... The son of the Holy Virgin, as they called Ceres, was
Osiris: he was born on the 25th of December. At his birth, Plutarch
says, that a voice was heard, saying, "On this day is born the
supreme Lord of the universe, the beneficent king Osiris."
[Emphases in original]
Plutarch's On Isis and Osiris does in fact quote a voice uttering something quite close to that line of dialogue on the occasion of Osiris's birth, although this has nothing to do with December 25th. It is from the same story in which we are told that Osiris was born in the extra (thirteenth) month of the year that was gambled into the calendar (see the section Osiris above).
"On the second of these days" (after Osiris's birth, i.e.), Plutarch says, "Haroeris was born whom they call Apollo, and some call him also the elder Horus... On the fourth day Isis was born... There is also a tradition that Osiris and Haroeris were sprung from Helios {Sol, in the Latin version; usually interpreted to be the Egyptian Ra}... Some say that Haroeris came from" the union of Osiris and Isis, who had married each other.
So Plutarch identifies Horus with Apollo, and not with Sol/Helios, whom he says is Horus's father in one version of the myth, although Higgins does not seem to care about such distinctions at this point, nor, for that matter does Dupuis before him, nor Doane afterwards. (The determination to cause any and all things on the subject to be about the sun is especially extreme when it comes to the etymology for Bethlehem, which in Hebrew means "House of Bread", and in Arabic "House of Meat", by whatever route it is that Higgins was able to arrive at "House of the Sun" for that instead.)
Ceres is in fact listed (by the Roman writer Hyginus) among the goddesses whom the constellation Virgo was supposed to be, and her Greek counterpart Demeter was indeed identified with the Egyptian Isis, but none of these deities, whether Roman, Greek or Egyptian, is ever a virgin mother, nor am I aware of any of them being named as the mother of Osiris.
The closest connection between December 25th and a son of Isis that I can find is when Plutarch says, once again in On Isis and Osiris, that Harpokrates, the Hellenised form of Horus the Younger, was given his birth by Isis "about the time of the winter solstice". To make this also about Osiris, Higgins dismisses the rather pertinent distinctions between them with the statement (Vol. 2, p. 102) that "They were the same Gods, in fact, only under different names."
Gold, Frankincense and Myrrh... or, Really, Any Assortment of Stuff
On p. 151 of Bible Myths and Their Parallels in Other Religions, Doane says:
Not only was Crishna adored by the shepherds and Magi, and received
with divine honors, but he was also presented with gifts. These gifts
were "sandal wood and perfumes." (Why not "frankincense and myrrh?")
[Italics in original]
Further on (p. 152), he adds that:
Buddha, as well as Crishna and Jesus, was presented with "costly
jewels and precious substances." (Why not gold and perfumes?)
Ensuring that Mithras doesn't get left out of his catalogue, he tells us (on the same page) that this god too, "at the time of his birth... was presented with gifts consisting of gold, frankincense and myrrh."
For the allegation about Krishna's gifts, he cites as his sources an encyclopaedia by Samuel Johnson as well as a book by Thomas Inman called Ancient Faiths Embodied in Ancient Names, Vol. 2 (both published about a decade previous to his own work). The Johnson reference simply narrates an episode in Krishna's adult life in which he is received as an honoured guest.
As for Inman, his Ancient Faiths has this to say (on p. 353):
No one can for a moment suppose that the Magi, who came to adore
Jesus and his mother, were Christians, making Christian offerings; the
context indeed paints them as Eastern Asiatics; and they are
represented as making the same oblations as they would to Mithra… in
their own district. Such offerings were made to the infant Christna
when in the arms of his mother Maia...
I cannot imagine any other reason for Inman to spell the name of "the infant" as Christna other than to insinuate some affinity between the otherwise completely unrelated words "Christ" and "Krishna." Also Maia (or rather Mâya) is actually the mother of the Buddha, and not of Krishna, but at this juncture the point seems to be to draw whatever parallels can be found, real or imagined, among these three figures. Inman's objective in this particular case seems to be the alliteration between the names Mary (Jesus' mother) and Maia/Mâya (attributed, falsely, with having given birth to Krishna).
At least the citation provided by Doane for the assertion regarding the Buddha comes closer to an undistorted representation of his source, which is Samuel Beal's 1875 book The Romantic Legend of Sâkya Buddha, in which a sage called Asita comes to visit the Buddha at his birth. The Buddha's father King Suddhôdhana (on p. 58) gives Asita "costly jewels and precious substances" but the sage "on his part... at once presented the gifts to the babe as an offering." The newborn's father and the sage then go back and forth a few times, the king trying to give the gifts back to Asita as originally intended but the sage insisting on offering them to the baby.
I haven't found any ancient narrative of Krishna's birth that says he received any sort of gift as a child, although in the Śrīmad Bhāgavatam he, upon being born, appears to his father Vasudeva bearing the dazzling attributes of Vishnu, wearing earrings sparkling with gems, sporting a full head of hair (covered with a helmet!), and generally "decorated with... armlets, bangles and other ornaments".
In the 1500s AD, a Bengali poet called Kavi Karnapura wrote a book entitled Ananda-Vṛndāvana-Campu, in which (per Chapter 2) the closest thing to the baby receiving gifts is that women bathe Krishna in some sort of ambrosia and then they anoint him with "fragrant sandalwood pulp" (apparently the same sort of treatment received after he grows up, as in Johnson's encyclopaedia).
Otherwise, to his mother, his appearance is so brilliant that his body looks to be made of jewels or lotus flowers. The villagers at his birthday celebration "decorated their cows with gold and jeweled ornaments. Then in great excitement they smeared them with oil, fresh butter, and turmeric paste." And his "relatives offered opulent cloth, jewelled ornaments, tāmbūla {betel leaves}, garlands, and sandalwood pulp to the guests." In Karnapura's book, these gifts, evidently quite different from the ones listed in Christ's birth story from Matthew's Gospel, are given to the guests, not the child.
As a bonus, on p. 96 of Anacalypsis 2, Higgins even grants us the notion that there was a "story of the three Magi, who, according to Plato, came from the East to offer gifts to Socrates at his birth, bringing gold, frankincense, and myrrh." The best we can hope to receive for a source for this claim (buried in Footnote 4 on the page) is: "This story of Plato's I cannot point out in his works, but I was told it by a most respectable clergyman at Cambridge." (The esteemed Cambridge clergyman is conspicuous in the fact that he remains unnamed.)
Notes
1. The work's full title is Anacalypsis, An Attempt to Draw Aside the Veil of the Saitic Isis; or An Inquiry into the Origin of Languages, Nations, and Religions, by the Late Godfrey Higgins [it was published 3 years after his death], London: Longman, Rees, Orme, Brown, Green, & Longman, Paternoster Row, 1836.
2. See p. 129 of Macrobius: The Saturnalia, Translated with an Introduction and Notes by Percival Vaughan Davies (Number LXXIX of the Records of Civilization Sources and Studies), New York & London, Columbia University Press, 1969.
3. My own translation of the original French, from p. 197 of Volume 2
4. See Beck's Review Article/Discussion "Merkelbach's Mithras" in the Vol. 41, No. 3, Autumn, 1987 issue of Phoenix, the journal of the Classical Association of Canada (p. 299, n. 12).
5. Forming part of Note 422 on p. 207 of Firmicus Maternus: The Error of the Pagan Religions, Translated & Annotated by Clarence A. Forbes (Newman Press, 1970, New York, NY & Paramus, NJ).
6. Ibid.
