My understanding is that the Christian idea of the underworld is more influenced by Greek mythology than Jewish. (In the New Testament, three terms are used for Hell, two Greek: Hades, Tartarus; and one Hebrew: Gehenna.)
Christianity itself may be said to be heavily influenced by both Greek Mythology (Dionysus/Persephone: Dying resurrected god; Heracles: deified son of Zeus; and even Achilles, who sacrifices himself for the community in an epic where the dignity of the gods is undermined and the dignity of humans is raised;) and ancient Greek philosophy (Platonism, Socrates' acceptance of his death sentence,
catharsis as fear&pity in a religious context, Elysium.) I'd even go so far as to say that segments of Judiasm are influenced by Greek thinking on humanism.
Although there are no demons per se in the Greek underworld, it is a very depressing place, where drinking blood is one of the few pleasures in an eternal limbo of wandering in darkness:
But when with vows and prayers I had made supplication to the tribes of the dead, I took the sheep and cut their throats over the pit, and the dark blood ran forth. Then there gathered from out of Erebus the spirits of those that are dead, brides, and unwedded youths, and toil-worn old men, and tender maidens with hearts yet new to sorrow, and many, too, that had been wounded with bronze-tipped spears, men slain in fight, wearing their blood-stained armour. These came thronging in crowds about the pit from every side, with a wondrous cry; and pale fear seized me.
Source: Odyssey XI
(I have a great attraction to early pagan religions where one dies and simply goes to "hell", with no talk of redemption. You see the same thing in Chinese thought, I think, before Buddhism, and ancestor worship is partly an effort to provide better material conditions in the afterlife, burning money and so forth.)
- The Greek underworld is a place of binding embodiments of ungovernable, destructive forces
Tartarus, a region of the underworld, is where the Titans are bound. Titans could be taken as a type of demon--they are often forces of nature, like the more abstract daimones, or "daemons", from which the word "demon" is derived. (Christianity, like all Abrahamic religions, casts nature spirits and lesser gods as demons, although in Christianity practiced in certain regions in history, such as the Carribbean, you also have certain pagan gods associated with saints or angels.)
- The binding and torture of Prometheus is an early precedent for eternal punishment
Prometheus' punishment is cast as unending, and he is only freed by the intervention of a son of God, in this case Heracles. (Deus is the most common form of Zeus in Greek literature, and the same word the used in Latin to connote the Abrahamic God.) Prometheus' punishment is also quite vivid, an element we definitely find in Christian mythology regarding the conception of hell.
- Tartarus and Hades are places of eternal punishment for mortals who have transgressed
Most famously, Sisyphus and Tantalus.
Greek mythology is filled with terrifying avatars of divine retribution, such as the Furies, who act in a demon-like manner to punish mankind. (The Furies were eventually pacified, but the principle is there, and many other instruments of divine vengeance abide.)
(Dīs Pater is the anti-sky father, or anti-Deus. The classical deity doesn't have the same degree of nefarious connotation as in Christianity, but is certainly a template. This deity, associated with Pluto, was also the god of wealth, as precious ores are delved from the earth, and as we all know, wealth was considered problematic in Christian thought, with avarice as a principal vice.)
I'm not an expert in Christian demonology, but in terms of casting Hades/Hell as a place of demons, it seems like a logical extension—if you are going to have eternal punishment for mankind, beyond just wandering around in the darkness forever, it makes sense that you would have agents for delivering very specific tortures.