Revealing mistake: When Robin is being pulled underwater by Ivy's plants he surfaces for air and gets pulled under again. It's obvious the film was just rewound. Then when he finaly gets free we see the shot that was rewound was the same shot used when he escapes. Really bad editing.
Continuity mistake: When Robin steps foot into Poison Ivy's lair she is laid out on her flower bed. If you look at the sleeves of her costume, the ends of the sleeves are fringed in a simple leaf like design. Robin spurns Poison Ivy's affections, and an enfuriated Ivy pushes him into the lily pad pool. She is then confronted by Batman, and a soon to arrive Batgirl. Now - look at the huge, obvious, black rubber "thorn" bracelets that suddenly appear on each of her wrists.
Revealing mistake: When Freeze is freezing the town, the cop car that crashed into the store has flexible icicles on the driver side door. Also when the city is being defrosted, when the cop comes out of the car, the icicles are very wobbly.
Continuity mistake: When Poison Ivy is talking to herself when she is leaving the observatory, you can notice that the policemen she pushed away have vanished in the background.
Other mistake: While fighting Mr. Freeze (and Bane) at the Gotham Observatory, Batman, Robin & Batgirl all have batsuits on, with silver bat (or robin) symbols on their chest. But while they are giving Alfred the cure to Stage 1 of McGregor's Syndrome, for some reason they have changed clothes before giving him the cure. Because now (other than the fact that they have all taken off their masks) Robin has on his original robin suit (with the red robin symbol across his chest) and Batgirl has on a gray t-shirt.
Continuity mistake: At the observatory, when Bruce gives Dr. Isley's (Poison Ivy) proposal back. She grabs it with the printed part toward her, but in the next shot, she's holding the paper with the painted part facing away from her.
Revealing mistake: After Arnie freezes Robin, then freezes the doorway, then wipes his hand on the ice to wave goodbye to our hero, you can see that the clear spot is there before he wipes it with his hand.
Factual error: Why is there a stellar observatory in the middle of a very large city, most of the stars would be drowned out by the city lights to the point of the observatory being useless. Also, the dome is very high up on top of a "building". Such buildings need to flex a little bit in the wind to keep them from breaking, this would make it impossible for the telescope to focus on anything for a timed exposure of film to make those neat pictures of galaxies and such.
Other mistake: When Mr. Freeze threatens to freeze all of Gotham City, he has a big snowglobe in his hand. He freezes the snowglobe, then shatters it in is hand. When he crushes it open with his hand, the inside of the snowglobe is empty. There is no water, frozen or unfrozen, that comes out of the snowglobe when it breaks.
Continuity mistake: When Dr. Freeze bends down to kiss his wife right before his accident, there is a tube of some sort behind her (in the distance), then as he kisses her the tube is gone.
Continuity mistake: The first confrontation with Mr. Freeze and his henchman involves them skating around a large ice covered area. The henchman are not using ice skates, but rollerblades. In the same scene we see ice skating blades pop out of Batman and Robin's boots, but when we actually see them skating, they are on rollerblades as well.
Continuity mistake: In Dr. Woodrue's lab, two thugs carry Antonio Diego over to an experiment table and chain him to it. The thug on Diego's left has a cowboy hat on. However, just as that thug is pulling Diego's mask over Diego's head a few shots later, the thug's cowboy hat is gone. This same thug also disappears from the aerial shots of the lab as Dr. Woodrue is exclaiming over his genius of turning Diego into Bane, a super soldier, with Venom. Later, when Bane breaks free of his restraints, the thug just shows up out of nowhere to try to hold Bane down.
Plot hole: When Freeze freezes the furnace after his plummet from the sky, why doesn't the pool of water that Batman puts Robin in freeze as well?
Other mistake: When Antonio Diego is being pumped up with Venom in Dr. Woodrue's lab, the tubes attached to the prisoner's head keep disappearing. It's in the shot where the camera pans from the foot of the exam table to Diego's (Bane's) right side.
Other mistake: Apparently, guessing Alfred's password also turns computer monitors into projectors. Before Barbara guesses the password, none of the light/information on the monitor reflects off her, but afterwards, she might as well be a projector screen.
Revealing mistake: In the opening heist at the Gotham Museum of Art, just after we see the large diamond soaring through the air and Batman incapacitating a goon, Robin skates over to a statue to remove its spear to use it as a vaulting pole. When he whacks the spear to remove it from its position, the icicles on the statue wobble, just like fake rubber icicles.