Continuity mistake: The location of The Penguin when he falls through the window and into the pool, changes. He moves much further away from the concrete platform. (01:45:50)
Visible crew/equipment: When the monkey jumps onto the cage in which Max Schrek is prisoner, you can see a cable clipped to its jacket it is wearing. (01:46:00)
Continuity mistake: When Catwoman pulls Max Schrek up out of the water, Max's hair is very wet, and is stuck down flat on his forehead. In the very next shot it is much drier and is wavier and stringy. (01:46:40)
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. (01:48:20)
Continuity mistake: When Selina removes her catwoman mask in front of Max, her hair is suddenly much more messy and tangled in the second shot where she says "Because he *is* Batman you moron." (01:49:05)
Continuity mistake: When Catwoman gives Max Schrek the taser-gun kiss, the black wire above them falls down at least four times. (01:50:20)
Continuity mistake: When Catwoman is electrocuting Max, when you see a view of the explosion from above, you can see Max walk off the set through the explosion. It is definitely him, not Catwoman!!! (check the clothing.) (01:50:30)
Visible crew/equipment: When the Penguin dies and the penguins slide him into the water you can see the slide-board he was laying on slip out ahead of him under the water. It is also very clear that the penguins aren't even touching him as they pull him into the water. (01:53:00)
Continuity mistake: When Alfred is drving Bruce in the car at the end of the film, the same car can be seen parked on the street, in different places, from different angles, about five or six times. (01:53:25)
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.
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 through 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.
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?
Continuity mistake: Selina grabs some stuffed animals and throws them in the sink, where she starts jamming a monkey down the garborator. We cut to her face, and when we cut back to the sink, all the other stuffed animals are gone, but we know she hasn't had a chance to shred them because she is still shredding the last of the monkey.
Revealing mistake: When Batman makes his first dramatic entrance to fight the clowns in his Batmobile, that clearly isn't snow being churned up by the wheels of his car.
Revealing mistake: When Penguin is controlling Batman's car, cars he is hitting throughout the scene are seen prematurely moving out of the way before he hits them.
Continuity mistake: In the scene where Catwoman and Batman fight each other you can see Catwoman's boots change several times from high heels to flat heels.
Continuity mistake: Just after Bruce has spoken to Alfred about him letting Vicky Vale into the Batcave in the past, we see him pulling his jacket on from behind. He puts his right arm in, then starts to reach behind himself to get his left arm in. The shot then cuts to seeing him from the front, and we see him reach behind himself again.
Continuity mistake: When the circus and Batman first make their appearance, one of the circus men grabs Selina Kyle and threatens Batman with a Stun Gun. After the man is knocked unconscious, Selina "zaps" the man. I don't know how many of your are familiar with stun guns but when they are activated on someone or something they don't spark or make a sound. The sound is only heard if held up in the air and is not in contact with anything. Yet when she zaps the man, there is a clear shocking sound.
Continuity mistake: When there's an overhead shot of the crowd in front of the Hall of Records, there is a person in bright red clothing amidst the people near the doors, pushing against a police officer, but in the next shot when Shreck is speaking to the press, the person is nowhere to be found and could not have exited the crowd in that short time.
Continuity mistake: When Penguin removes the manhole cover above him right before the Mayor's baby is stolen, he places it right next to the edge of the manhole, but when the baby snatcher jumps in, it is further away.
