Queen Dido
The legendary Carthaginian Queen Dido kills herself in despair after her lover Aeneas, son of Aphrodite, abandons her.
Tristan & Isolde
Joseph Campbell wrote about these two, emphasizing the unification of divine and romantic love (he calls it "Roma" vs. "Amor", Roma referencing the seat of the Catholic church in Rome.) The legend provides the template for Romeo & Juliet, the background for Malory's Le Morte d'Arthur, and is the subject of one of Wagner's operas, quoted in The Wasteland.
Tristan was now too weak to keep his watch from the cliff of the Penmarks, and for many long days, within walls, far from the shore, he had mourned for Iseult because she did not come. Dolorous and alone, he mourned and sighed in restlessness: he was near death from desire.
At last the wind freshened and the white sail showed. Then it was that [Tristan's wife] took her vengeance [telling him his lover would not come to heal him]. And Tristan turned him to the wall, and said: “I cannot keep this life of mine any longer.” He said three times: “Iseult, my friend.” And in saying it the fourth time, he died.
...
Iseult came in and said [to her]: “Lady, rise and let me come by him; I have more right to mourn him than have you—believe me. I loved him more.”
And when [Iseult] had turned to the east and prayed God, she moved the body a little and lay down by the dead man, beside her friend. She kissed his mouth and his face, and clasped him closely; and so gave up her soul, and died beside him of grief for her lover.
“Le Roman de Tristan et Iseut,” by M. Joseph Bédier (Gutenberg.org)
After their deaths, a rose briar grows up above their tombs, entwining, as a symbol of their immortal love. This incredibly influential legend is believed itself to have been influenced by Pyramus and Thisbe.
Pyramus and Thisbe
In Metamorphoses IV. 55-166 you will find the story of Pyramus and Thisbe, an early example of star-crossed lovers. A brief summary of the plot, from Wikipedia:
In Ovid's Metamorphoses, Pyramus and Thisbe are two lovers in the city of Babylon who occupy connected houses/walls, forbidden by their parents to be wed, because of their parents' rivalry. Through a crack in one of the walls, they whisper their love for each other. They arrange to meet near Ninus' tomb under a mulberry tree and state their feelings for each other. Thisbe arrives first, but upon seeing a lioness with a mouth bloody from a recent kill, she flees, leaving behind her veil. When Pyramus arrives he is horrified at the sight of Thisbe's veil in which the lioness had torn and left traces of blood behind, as well as its tracks. Assuming that a wild beast has killed her, Pyramus kills himself, falling on his sword, a typical Babylonian way to commit suicide, and in turn splashing blood on the white mulberry leaves. Pyramus' blood stains the white mulberry fruits, turning them dark. Thisbe returns, eager to tell Pyramus what had happened to her, but she finds Pyramus' dead body under the shade of the mulberry tree. Thisbe, after a brief period of mourning, stabs herself with the same sword. In the end, the gods listen to Thisbe's lament, and forever change the colour of the mulberry fruits into the stained colour to honour the forbidden love.
Wikipedia contributors. (2019, May 3). Pyramus and Thisbe. In Wikipedia, The Free Encyclopedia. Retrieved 20:04, May 4, 2019, from https://en.wikipedia.org/w/index.php?title=Pyramus_and_Thisbe&oldid=895386980
Hero and Leander
A brief summary of the story from Britannica:
Hero and Leander, two lovers celebrated in Greek legend. Hero, virgin priestess of Aphrodite at Sestos, was seen at a festival by Leander of Abydos; they fell in love, and he swam the Hellespont at night to visit her, guided by a light from her tower. One stormy night the light was extinguished, and Leander was drowned; Hero, seeing his body, drowned herself likewise.
The story is preserved in Musaeus, Ovid, and elsewhere. It was also adapted by later poets, such as Christopher Marlowe (Hero and Leander), and alluded to by Lord Byron (The Bride of Abydos).
The Editors of Encyclopaedia Britannica. Hero and Leander. Encyclopædia Britannica. Retrieved May 07, 2019, from
https://www.britannica.com/topic/Hero-Greek-mythology
Phaedra and Hippolytus
The love of Phaedra for Hippolytus is unrequited, but it ends in death(s) too:
Though married to Theseus, Phaedra fell in love with Hippolytus,
Theseus's son by another woman (born to either Hippolyta, queen of the
Amazons, or Antiope, her sister). Hippolytus rejected her.
In revenge, Phaedra wrote Theseus a letter that claimed Hippolytus had
raped her. Theseus believed her and cursed Hippolytus with one of the
three curses he had received from Poseidon. As a result,
Hippolytus's horses were frightened by a sea monster and dragged their
rider to his death.
In another version, after Phaedra told Theseus that Hippolytus had
raped her, Theseus killed his son, and Phaedra then committed suicide
out of guilt, for she had not intended Hippolytus to die. Artemis
later told Theseus the truth.
Wikipedia contributors. (2019, May 7). Phaedra (mythology). In Wikipedia, The Free Encyclopedia. Retrieved 15:04, May 7, 2019, from https://en.wikipedia.org/wiki/Phaedra_(mythology)
Oenone and Paris
Oenone was a mountain nymph and the wife of Paris, until the Trojan prince abandoned her for Helen. After the fall of Troy the once lovers met again:
Mortally wounded by Philoctetes' arrow, he begged Oenone to heal him
with her herbal arts, but she refused and cast him out with scorn,
to return to Helen's bed, and Paris died on the lower slopes of Ida.
Then, overcome with remorse, Oenone, the one whole-hearted mourner of
Paris, threw herself onto his burning funeral pyre, which the
shepherds had raised. A fragment of Bacchylides suggests that she
threw herself off a cliff, in Bibliotheke it is noted "when she
found him dead she hanged herself," and Lycophron imagined her
hurtling head first from the towering walls of Troy. Her tragic story
makes one of the Love Romances of Parthenius of Nicaea.
Wikipedia contributors. (2019, May 8). Oenone. In Wikipedia, The Free Encyclopedia. Retrieved 15:04, May 8, 2019, from https://en.wikipedia.org/wiki/Oenone