Plot hole: When The Penguin is controlling the Batmobile, Batman punches through the floor to take off the transmitter. We still see a video feed of The Penguin. Why? Batman pulled off the transmitter, so there is no reason for there to still be a video feed.
Suggested correction: We see the Red Triangle gang spends a while fiddling with the Batmobile's workings before installing the transmitter. It's likely they made multiple 'modifications', thus the transmitter was for controlling the car's engine and steering, and the video feed was made possible by another, separate means.
Suggested correction: I concur with the other correction. There are multiple shots showing the gang working on the car and doing other things to it beyond just putting on the transmitter. (You see them playing with wires, moving parts around, etc.) Hacking into the computer/video-feed so Oswald could taunt Batman is likely among the other things they are doing. There's nothing into the movie that suggests taking off the transmitter should (or even could) interrupt the video-feed. The fact that Batman has to punch the screen to get rid of the Penguin's image is another point to the fact that the transmitter itself had nothing to do with the video feed.
Audio problem: The Batmobile is supposed to be powered by a Jet engine. The sound in the sequel is dubbed rather flawlessly except for two brief instances when you can hear the 8 cylinder piston engine.
Suggested correction: Can you be more specific as to what point in the film these errors occur?
Revealing mistake: When the clown grabs the Mayors baby, it is obviously a doll. (00:33:10)
Suggested correction: There's no shot of the baby's face or body that reveals he's a doll. When he's abducted it's a real baby we see. The clown then does some flips and quick movements where the baby's face or body aren't seen at all. (We "know" he is a doll, of course, but there's no visual proof of it).
