No, Bottom is blissfully unaware of his strange transformation. He cheerfully goes about this business without the slightest realization that the mischievous Puck has given him the head of a donkey.
Strange as it may seem, there are certain distinct advantages to Bottom's unusual condition. Unique among the characters on the stage, he can move back and forth freely between the human and fairy worlds. This allows Shakespeare to further blur the distinction between these two radically different realms, between what is real and what is mystical.
It is only because Bottom is unaware of what's happened to his head that he's able to transgress established conventions. Thanks to Puck's crafty magic, Bottom is no longer a humble weaver: he has taken on the persona of a fairy queen's lover. If he ever became aware of his donkey's head, the spell would be broken, and the play wouldn't be quite as effective in its challenging of accepted social norms.