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.
Continuity mistake: During Barbara Gordon's flashback to when Robin went missing, when Batman and Batgirl enter Arkham Asylum, Harley Quinn alerts the Joker that they have company. The Joker is seen with a pipe in his mouth. He takes the pipe into his right hand as he starts to come down the stairs. In the very next shot, the pipe is missing. (00:37:20)
Deliberate mistake: The device that rises out of the Jolly Jack factory and summons the laser beam shatters the decorative head, appearing to do so for the first time. However, we have already seen the destructive beam in action, meaning that device must already have emerged. This was acknowledged on the commentary by the filmmakers who wanted to show the visual metaphor of the Joker driving the young Tim Drake insane.