The first allusion is in the epigraph, which is taken from Dante's Divine Comedy. In English it reads:
If I thought my answer were to one who could ever return to the world, this flame would move no more; but since no one has ever returned alive from this depth, if what I hear be true, without fear of infamy I answer you.
Prufrock has to know the answer to his question to the woman as a condition of asking it, or he needs not to care what the answer will be. Neither of these is true, so he can't bring himself to ask it.
Another allusion is to "time for all the works and days of hands/That lift and drop a question on your plate..." The allusion here is to a poem by the classical poet Hesiod, about an ancient Greek farmer that is encouraging his brother to work as hard as he is. Prufrock imagines that other hands are working harder than him, a condition that necessitates his asking the question. But he imagines that he has time for tea first.
Later, Prufrock imagines himself as John the Baptist:
But though I have wept and fasted, wept and prayed/Though I have seen my head (grown slightly bald) brought in upon a platter/I am no prophet, and here's no great matter...
John the Baptist's head was presented on a platter to Herod's wife. He is associating the weeping and fasting that biblical prophets engaged in with his own hemming and hawing over approaching the woman and asking the question.
Finally, and perhaps most famously, there is a reference to Hamlet, when Prufrock exclaims, "No! I am not Prince Hamlet; nor was meant to be..." This is an allusion to the famous "to be or not to be" soliloquy, and Prufrock is comparing his own inner dialogue with this famously tortured passage.
Generally, Eliot seems to be making these allusions in an effort to show that his protagonist is trying to lend an epic, even heroic air to his tortured unwillingness to approach the woman.