Deliberate mistake: When Barbara Wilson/Batgirl first arrives she is standing on a ledge beside the front door when Robin answers the door. She then exclaims, "Oh. Uncle Alfred." However she could not possibly have seen him, not just because of her angle, but also the fact it is after that statement that he first comes into view.
Deliberate mistake: At the beginning when Batman and Robin are facing Freeze, Robin arrives and crashes through the wall with his motorcycle and leaves a shape in the wall that a motorcycle could possibly make. But, the next shot after Batman looks where Robin is still flying and about to land on his motorcycle, there is a huge bat shape in the broken wall. Obviously done for campy effect, but still a mistake regardless.
Deliberate mistake: During the opening battle scene, Mr. Freeze's gun gets stuck on top of a statue. Mr. Freeze then throws a chubby security guard up straight up into the air to knock his gun down. But it's completely wrong and defies the laws of physics. The security guard bumps the gun from the front, causing it to fall FORWARD and STRAIGHT DOWN, when in reality, the angle from which the guard smacked it in the front should have knocked the gun backwards even more. This reveals that the gun was just pulled down by a wire or was pushed forward from behind, as it completely defies the laws of physics and action/reaction.