Prometheus was a Greek titan who, according to myth, gave us fire (among other things). Like with many Greek mythological figures, there are many different legends about him. The most definitive work about him is the book "Prometheus Bound" by Aeschylus.
Here are the claims given by critics:
1. He descended from Heaven as God incarnate to save mankind.
First of all, Prometheus wasn't 'God'. The Greek equivalent of the Judeo-Christian God was Zeus. There were also many lesser gods in Greek religion, and Prometheus wasn't even a lesser god. He was a titan. Neither was Prometheus aligned with God/Zeus as Jesus was, but was frequently in conflict with him.
Prometheus did, however, save mankind, but not in the way Jesus did. Prometheus did not save us from sin or damnation in the afterlife, but had stopped Zeus from destroying mankind.
2. Was born of a god and a human virgin on December 25th.
Neither of his parents were gods, humans or virgins. They were both titans. His father was Iapetos, his mother Klymene. And Klymene was not a virgin. According to one version (Theogany 508-510), "Iapetos took Klymene, the light-stepping daughter of Okeanos, to be his wife, and mounted into the same bed with her." And there's nothing about his being born on 12/25.
3. He had a friend named 'Patraeus' (Peter), a fisherman, who deserted him.
The critics here are referring back to Oceanus, with one of them stating that the names Oceanus and Patraeus are interchangable. The critic fails to explain how they're interchangable, and there's no rules in Greek language or name systems which would suggest that they are. Oceanus was a fellow titan associated with rivers, though there is no reference to his being a fisherman or catching any fish. In "Prometheus Bound", he appears briefly, offering to intercede with Zeus over a disagreement between Zeus and Prometheus. Prometheus sends Oceanus away, afraid that his intercession would put Oceanus at risk. This hardly qualifies as "deserting" Prometheus, at least not in any of the sense that Peter deserted Jesus.
4. He was crucified, suffered, and rose from the dead.
Well, he suffered, at least. Prometheus (like all titans) were physically immortal and could not die. In Greek legend, Zeus sentences Prometheus to eternal torment, shackling him to the side of a rocky crag in the Caucasus Mountains. Zeus' pet eagle (or, in some versions, a vulture) attacks Prometheus every day, giving him terrible wounds that heal at night. Prometheus is eventually rescued by Herakles, who kills the bird. This incident seems to be the 'crucifixion' the critics refer to. But since there's no crucifix, no death, and no resurrection, this hardly compares to what happened to Jesus.
5. Died for mankind's sins.
The source for this claim appears to be Robert Taylor's "Diegesis". Taylor quotes Robert Potter's 1779 translation of Aeschylsus's "Prometheus Bound" as saying: "Lo, streaming from the fatal tree, His all-atoning blood! Is this the Infinite? 'Tis he - Prometheus, and a God! Well might the sun in darkness hide, And veil his glories in, When God, the great Prometheus, died, For man, the creature's sin." The problem is that this passage appears nowhere in Potter's translation, or any other. You can read Potter's translation for yourself here: Prometheus Bound. The words "died" and "sin" don't even appear anywhere in the translation. According to J.P. Holding (page here), Taylor may have been confused, thinking of a hymn by Isaac Watts, which says, in part, "When Christ, the mighty Maker, died, For man, the creature's sin." Even Acharya S admits that Taylor's quote is false on her website.
6. He was called the Logos or the Word.
Yes, but not commonly, and not with the same meaning at which Jesus is called 'Logos'. Christians believe that Jesus is the Word (Logos) of God manifested into the flesh. The Greek mythologian Plutarch called Prometheus "reason" (also Logos), but did not mean in the sense of Prometheus being the Word of Zeus.
7. He was betrayed by someone close to him.
This "comparison" was sent to me in an E-mail, but the writer wasn't specific about who the friend was (I tried e-mailing the writer back, but my mail was blocked). Hephaestus and Zeus could both be said to have betrayed Prometheus one way or the other. Zeus was the one who sentenced Prometheus to be chained to the rock, despite the fact that Prometheus had earlier helped Zeus defeat the titans and they'd been friends, so this certainly qualifies as a betrayal. Prometheus and Hephaestus had also been friends, and Hephaestus was the one who carried out Zeus' sentence, so this is a betrayal as well. But neither of these are the type of betrayal that Judas carried out against Jesus in the Gospels, accepting money to reveal the location of Jesus to the authorities.
8. When Io visited Prometheus, as he was chained to the rock, she brought along a swarm of gadflies which darkened the sky, just as the sky darkened while Jesus was on the cross.
True, except for where it compares to Jesus. Io was surrounded by a swarm of gadflies when she visited a bound Prometheus, but nothing in the story suggests that the swarm was so great that it darkened the sky (in fact, in some versions of the story, it's a single gadfly which bothers her). The gadflies were a punishment from Zeus against Io alone, so the idea that the swarm was sky-darkening doesn't make much sense, since that would punish everyone around her for miles. The story suggests that the swarm was only in her immediate vicinity.