That's quite a hard question. For my money, most of the paradoxes that spring up in the strange, disturbing final acts of this play revolve around Juliet's supposed death, and around a key Elizabethan pun on the word "death". "To die", to the Elizabethans, didn't just mean "to stop living", but also "to have an orgasm".
One paradox comes, I think, when Romeo hears of Juliet's death from Balthasar. Here's what he says:
Well, Juliet, I will lie with thee to-night.
What Romeo means is that he will lie dead with her tonight, committing suicide next to her dead body. Of course, Romeo will lie with her - in that, she will be alive when he physically lies next to her. And in an odd sort of way, he will "lie with her" (which can also mean "sleep with her") - as the two of them will "die" (orgasm) next to each other. Paradox after paradox there.
And, as the two of them die, look at what they say:
O true apothecary!
Thy drugs are quick. Thus with a kiss I die.
O happy dagger!
This is thy sheath; there rust, and let me die.
Both of them, lying next to each other, "die" on the word "death". And both of their deaths have a sort of sexual context - Romeo dies with a kiss, and Juliet by putting a dagger (phallic symbol) into her own body, a "sheath" (an Elizabethan word for the female genitals). So Romeo and Juliet die (die) and die (orgasm) at the same time. To have the beginning of life and the end of it at the same time can only be a paradox.