Other mistake: When searching the buoy for clues, Batman goes to take out an obvious screw that was supposed to hold on the panel hiding the projector. The screw is located in the top of a seam where two panels were riveted. After removal, the panel is removed and you can see the seam again that would have been hidden by the panel, as it was tall enough to fit the side from the walkway to the top. Also, the lenses on the projector were obviously fake. Any projector with lenses that badly chipped and faded would have been repaired or replaced. In addition, Batman is supposedly loosening the screw, but was turning the screw to the right, which would be tightening it.
Continuity mistake: At the airport, the Batcopter is ready to fly. As it ascends, look at the front of the bottom side. The only thing there is a standard search light. Later, when over the yacht, Batman drops the Bat-ladder, which appears to be attached to the bottom front next to the light, but wasn't there on take-off. The class of chopper used didn't have room for a hidden compartment.
Other mistake: After opening the abandoned elevator, the Penguin tells his cohorts to hold their breaths because of lingering gas, and there was no real sign of it. However, before the elevator, there was colored gas all over, and he walked through with no ill effects, and held his breath with the others.
Continuity mistake: In the beginning, after Batman and Robin are in the Batcopter, in a close-up shot, you see Robin with the Bat-camera next to him, when there wasn't anything in the cockpit with them when they took off. After he raises it to get a reading and take pictures, a wide angle shot shows them in the chopper with no camera, and then another close-up, and the camera's back.
Continuity mistake: When Batman and Robin are circling in the Batboat over the submarine with Robin shooting at it inside the circle, the boat is shown circling in the same tight circle while underwater shots of the submarine show it moving forward under the water. The sub's position would have been changing so the Batboat would have had to change position with it.