The scene where Romeo is wangst-ing to Benvolio about his breakup with Rosaline, and his overdramatic soliloquy gets interrupted by the entrance of a Capulet servant. Particularly funny because it seems like something that would happen in a parody of Romeo and Juliet. Shakespeare wasn't above poking fun at his character's melodrama (although the real drama for Romeo hasn't set in yet at this point in the play).
Not mad, but bound more than a madman is,
Shut in a prison, kept without my food,
Whipped and tormented and...god-den, good fellow.
The scene where the Nurse takes forever to get to the point, driving Juliet completely up the wall.
Juliet: How art thou out of breath, when thou hast breath/To say to me that thou art out of breath?
Especially when said in the stereotypical annoyed-teenage-girl voice.
Pretty much any of the Nurse's filthy jokes count too.
The Nurse tells a story about Juliet as a toddler, indicating she and her late husband shared the same dirty sense of humour:
For even the day before, she broke her brow,
And then my husband — God be with his soul!
He was a merry man — took up the child:
"Yea," quoth he, "dost thou fall upon thy face?
Thou wilt fall backward when thou hast more wit;
Wilt thou not, Jule?" and, by my holidam,
The pretty wretch left crying and said "Ay."
Before that, Mercutio gleefully mocking the Nurse (in some productions, in tandem with Benvolio) and her reaction.
(Enter Nurse and PETER.)
Mercutio: A sail, a sail!
Benvolio: Two, two; a shirt and a smock.
Nurse: My fan, Peter.
Mercutio: Good Peter, to hide her face; for her fan's the fairer face.
Depending on how it's performed, this line:
Nurse: Is it good e'en?
Mercution: 'Tis no less, I tell you, for the bawdy hand of the dial is now upon the *prick* of noon.
Mercutio has the distinction of being funny even when he's dying. "Ask for me tomorrow, and you shall find me a grave man." And all his other incredibly filthy jokes count too.
Peter threatening to kill the musicians if they don't play him a song.
Depending on your sense of humor, the ending is hilarious.
Seriously guys, there's like a dick joke a scene, especially in the early part of the show, when Sampson is telling Gregory of how well-endowed he is.
Sampson: I am a pretty piece of flesh!
Gregory: 'Tis well thou art not fish; if thou hadst, thou hadst been poor John.
Note that poor John is a tiny, pathetic dried fish