Deliberate mistake: You might be wondering where Terry suddenly comes up with Jolly Jack all of the sudden after escaping the lab. In the commentary, they say that the Joker, in his transmission to Terry in the lab, was supposed to be eating out of a candy jar with "Jolly Jack" on it, but couldn't have it due to time constraints.
Audio problem: When Tim enters Gordon's office, you hear Tim greeting Barbara, but it is Barbara's mouth that is moving. When Barbara replies, her mouth isn't moving.
Continuity mistake: When Bruce plays the clip of the Joker appearing at the rooftop party, the audio clip that plays with it is not the same as when the Joker actually said it during the attack. The clip seems a bit more subdued than the actual voice track.
Trivia: In the original version, Tim does not throw the gun down. He shoots it at the Joker, who dies when he's impaled with a flag shooting out the gun reading, "Bang".
Trivia: When the film was initially released, there was still ongoing backlash against violence in media (particularly media aimed at children/teens) as a result of the Columbine school shooting. Warner Bros. got cold feet and had the film edited and new alternate scenes written and animated to cut down on the violence on-screen. (Most notably in the "Death of Joker" flashback which not only had Joker dying in a different manner, but had a sequence cut showing Joker stabbing Batman in his leg, giving him his limp) Warner Bros. Initially did not wish to release the original version of the film, but gave in after fans petitioned to see the original intended version.
Batman: He's tough. Any suggestions, boss?
Bruce Wayne: Joker's vain and likes to talk, he'll try to distract you, but don't listen. Block it out and power on through.
Batman: Wait... I like to talk, too.