Revealing mistake: Batman drives out of the batcave alone. The first bit of stock footage of the car leaving the cave does indeed show him alone in the batmobile. But the very next stock shot of the car speeding down the dirt road shows us Robin sitting in the passenger seat.
Continuity mistake: When Batman and Robin are in the museum speaking with the director, Batman's chest emblem is high enough on his chest that it nearly touches the bottom edge of his cowl. In the next shot, when he leaves the museum, the emblem has moved down several inches.
Add timeJeff Swanson
Continuity mistake: Molly is about to disguise herself as Robin. But the wig she's holding isn't even remotely the same color as Robin's hair. Of course, as soon as she puts it on and transforms herself, the wig darkens and becomes Robin's actual hair.
Continuity mistake: While the unconscious Robin is being carried away from the Batmobile by one of Riddler's thugs, his cape loosens and appears to completely fall off just before they exit the frame. In the next shot, Robin is still being carried away but his cape is securely back in place.
Add timeJeff Swanson
Continuity mistake: When Batman starts the Bat-Turn, the parachutes are shown deploying on a 6-lane major roadway (3 lanes in each direction) in a very open area, but after completing the 180-degree turn, the car is shown driving away from the crumpled chutes on a much narrower side street among tall buildings.
Deliberate mistake: Ignoring that its very unlikely all of Jill St. John's copious red hair would fit under a wig that passes for Robin's hair, it's impossible that it would emerge from under that wig looking as full and perfectly styled as it does when she's in the Batcave.
Factual error: Though he's a bastion of knowledge and a member of the museum board, Bruce Wayne completely mangles Egyptian history - twice. He says that King Tut ruled in the 14th Dynasty, and that the museum's mummy was a 14th Dynasty king who ruled in 1500 BC. Wrong on both counts. Tut reigned from 1336-1349 BC. The 14th Dynasty ran from 1773-1650 BC. Wayne would have known that - and so would his well-educated colleagues, none of whom points out the error.
Other mistake: Riddler says that his "universal wax solvent" can eat through anything. Which begs the question, just what is that vat he cooks the stuff up in made of? We see the wax burn through solid steel vault doors. So it also ought to be eating through the vat, the floor and the bedrock, and sizzling its way down to the Earth's core.