Continuity mistake: As Lex and Joker are talking on the bridge between two towers of Arkham Asylum, Joker is shown standing in the middle of the bridge. But when the guy on the intercom announces there is an escape going on, the camera changes angles and suddenly Joker is right next to the edge.
Continuity mistake: After his missiles fail to destroy Batman's water vehicle, the Joker then throws out numerous items (a carrot, a photo and an inflatable monkey) to find his electric pies which then land on the ground. His electric pies also fail to destroy the water vehicle so he steps out of his vehicle and begins walking over to it. After this there are 3 different shots where he says the line "Where does he get these unbreakable toys?" In the second shot during this line, the stuff he threw out of the vehicle to get the electric pies has vanished without a trace.
Deliberate mistake: When Lex busts Joker out of Arkham Asylum, he uses his Deconstructor gun on the LEGO wall. It takes it about 8 seconds to pull the LEGO apart for Joker to get out. The same amount of time is taken when he demonstrates it to Joker and frees Penguin. Yet when Joker grabs it and uses it to free Catwoman and Two-Face, that time is cut in half. And then yet again when they are leaving on the chopper, Joker fires it 3 more times, freeing Poison Ivy, Harley Quin, and The Riddler. It barely takes a single second on each of those.
Continuity mistake: When Batman and the Joker land at the amusement park, the Joker takes out several missiles and says to Batman, "You like missiles Batman? Because I think they're going to like you" and shoots them at Batman's water vehicle. Within the first second of shooting, the missiles already make it more than halfway to Batman. After this, Batman places a cloaking device over his water vehicle to shield it from the missiles. This takes 4 seconds to complete, but the missiles somehow don't cover the remaining distance in that time.
Continuity mistake: When the Joker crashes the award ceremony, he jumps onto Lex's table and kicks a lamp off the table and it lands on the floor. After kicking it onto the floor, you can hear a smashing sound coming from the lamp. But when Riddler, Penguin, Two-Face, Catwoman and Harley Quinn sit down on the same table Lex is at, the "smashed" lamp is now fully intact.
Continuity mistake: When Lex breaks the wall of Arkham to let Joker out, the view from inside the room shows him just a few feet away from the hole. But in the wide shot when Joker is walking out, Lex is a much greater distance away from it. The distance is too great for him to have just decided to step back for some reason.
Continuity mistake: After Batman hits Joker with a net and pins him to the ground, he gets out of his jet and hops down. There is a chicken leg next to the jet from Joker's previous attack. But when Batman is getting back in while Superman and Robin are talking, the chicken leg has disappeared.
Continuity mistake: After taking out Joker, a police officer thanks Batman for the help. Robin says he was a part of this too. There is a police car just beside Batman, and the cop is right in front of him. It then switches to a high angle before Robin finishes his statement - suddenly the cop is gone, and the police car has changed positions, now driving away from Batman.
Continuity mistake: When Lex and the Joker exit Ace Chemicals, there are three canisters next to the door. Lex opens the door 90 degrees which isn't enough to knock over the three canisters next to the door. But when Batman and Robin enter Ace Chemicals later, Batman opens the door less wide than Lex, but somehow knocks over the canisters.
Continuity mistake: At Arkham Asylum, Bane says "Not this time". The next shot shows the ground Batman and Robin are currently standing on rattling around. We can see the edges of the tiles, so we know they're rectangular. In the next shot the tiles are now square. Don't confuse them with the tiles to the left of Batman and Robin.
Continuity mistake: Bruce goes onto the stage to accept his award and starts a speech. In the reverse angle where Bruce says "I do what little I can", we can see a woman sitting next to Lex is looking at some annoyed guy also sitting next to her. In the next shot, she's suddenly looking straight forward at Bruce.
Continuity mistake: When the award ceremony begins, Bruce exits his limo and places his arms around two girls' shoulders. His arms are stretched out around their shoulder area plus you can see part of his arms just past the girls' necks so his arms are definitely around their shoulders. Yet as the camera is behind Bruce, his arms are now lower down around the girls' backs.
Continuity mistake: In the opening scene, Superman and Batman land in front of LexCorp. In the wide angle, there is lots of traffic directly in front of the building. But when it cuts to the reverse angle, there's barely any traffic there. The LexCorp building does rotate, but only does it slowly and the traffic could not have disappeared that quickly.