Continuity mistake: Just before Batman takes his mask off at the end, the black makeup around his eyes vanishes while the mask is still on.
Continuity mistake: In Schreck's department store, when Catwoman is doing her cartwheels, her heels collapse.
Revealing mistake: In one scene, the Penguin is making a "bat" shadow puppet on the ceiling. But there's no light behind his hands. If there's no light, there's no shadow.
Deliberate mistake: When the penguin goes into the cemetery, he bumps a tombstone and it moves. Almost certainly a tribute to legendarily poor director Ed Wood, who Tim Burton later directed a biopic of - this exact thing happened in one of his films.
Revealing mistake: When the Penguin drives the Batmobile remotely, the cars it smashes are damaged just before it hits them.
Continuity mistake: In Batman's first battle with the Red Triangle Circus gang he manages to escape from a fire eater by turning his batmobile around and burning the guy with the rear exhaust jet. He does this by flicking a switch (outlined in caution symbols) next to the steering wheel. When the Penguin takes Batman for a ride there is a television screen where the switch is supposed to be. When Batman blows the Batmobile apart to get though a narrow alley the switch is back in place. Not only that but the switch seems to control just about every major system on the car including the lights, shield cover, gun turrets (in the first movie), canopy etc.