When Andrea first encounters Batman in her apartment and he asks her about her father, he hands her a photograph with his bare hand. In the very next shot his glove is back on.
When Andrea comes to Bruce's house the first time and Bruce is practicing jiu jitsu, he trips her. When she hits the ground she is wearing a sweater, but previously she was wearing a sleeveless blouse.
During Batman and Joker's fight, Batman kicks out one of Joker's teeth, later on when Joker puts on his jet pack and starts to take off he has all of his teeth, but in the next shot his tooth is missing again.
When Batman questions Andrea about her father, he takes out the framed photo he took from Sal Valestra's house, but then shows her a frameless picture.
Just before Batman comes to question Arthur Reeves in his hospital room, the doctor and orderlies tell him to rest and clearly slam the door. In the next shot of the room's door, it is half open. Then, immediately after Batman is seen, it is fully closed.
While jiu jitsu does have strikes, it is primarily a grappling art that is normally practiced with two people, and Bruce is exclusively seen performing kicks, punches and blocks by himself, which makes it more likely that he is practicing something like karate instead. When Andrea throws him with a move from her self-defense class, that seems closer to jiu jitsu than what he is doing.
The construction site Batman hides in has bombs labeled on it. However, construction crews don't use explosives when building something.
The Gotham City Police try to shoot Batman when he is in a cloud of tear gas, and high above them. In real life, law enforcement are not allowed to shoot at a target if they can't see them, and if they don't know if there is a potential hazard.
During the opening sequence, the camera pulls back dramatically through the myriad buildings of Gotham City. As this occurs, the camera actually passes through several buildings - a common 3D animation problem when rendering. As a result, a few buildings "pop" into view out of nowhere.
When Andrea is being sucked into the fan at the Gotham World's Fair, Batman rides in on his motorcycle. He turns into the fan to rescue her, but his cape is flapping in the opposite direction to the fans sucking.
After Batman has left Andrea's apartment she drops a glass. We hear the sound of the glass shattering, but when the camera zooms out we see the glass is still in one piece.