Well, of course the title of the play suggests that the whole event is, in itself, a dream. And at the end, Puck says to the audience:
If we shadows have offended
Think but this and all is mended --
That you have but slumbered here
While these visions did appear.
And this weak and idle theme
No more yielding but a dream.
While it seems in these lines that Puck is apologizing to the audience if they, perchance, did not like the play, he is also gently chiding those who might take it all too seriously. "This is a play, people," he might be saying. "Lighten up! Enjoy! It isn't intended to be any more logical or real than a dream you might have."
Hermia is a character who actually falls asleep and has a dream in the play. When she lies down beside her true love to sleep in the forest, she awakens to find that he has "fallen in love" with Helena. Hermia screams for Lysander's help to "pluck this serpent from my breast," which indicates that she has dreamed that a snake has come to attack her, a common dream image of treachery, foreshadowing her assumption that Helena has "come by night" and "stolen my love's heart from him."
Bottom and Titania both claim to have dreamed their drug-induced romance, but while Titania sees her dream of loving an "ass" as a nightmare, Bottom seems inspired by the events he considers to have been a dream. He says that he will have a play written about it, of course casting himself as the star, called "Bottom's Dream." In this idea, he suggests somewhat the process of the playwright. Dreams could be the source of the playwright's ideas and plots.
Of course, there are many more connections between dreams and the play to be made. Please follow the links below for more information on "dreams" in Midsummer.