Great answer above!
I think another aspect that relates to dreams is that ultimately, everyone ends up with who they are meant to be with, which doesn't happen often in real life, so this could be dream-like, as well.
The things that happen to many of the characters are fantastical, i.e., they could not really happen. Bottom gets a donkey's head (and doesn't realize it). Demetrius and Lysander fall in love with whomever each sees first upon awaking. Titania, the queen of the fairies, also falls in love at first sight with Bottom, a common mortal. All of these events are unreal, or dreamlike. Plus, they all happen within the course of one night. Puck says, in his final speech in Act 5, sc. 1, that if there is anything that the audience didn't like about the play, to just look upon the play as a dream - something that never really happened.