The Southern Air Temple - S1-E3
Other mistake: When the gang is in the Hall of Statues of Past Avatars, Katara points out the avatar cycle pattern of statues and Aang zeroes in on Roku's statue. The earthbender avatar statue next to Roku is of an unknown man with a sword, not Kiyoshi. (00:12:50 - 00:13:35)
Other mistake: When Azula attacks The Western Air Temple, there is a close up shot of her when she says that she is about to become an only child. Pay attention to the smoke coming from the airship behind her, especially above her head: it doesn't move. Even if this was remotely possible, it does move when the camera zooms out, making it a mistake nonetheless.
Bato of the Water Tribe - S1-E15
Other mistake: When Katara is behind Sokka and he is speaking she leans back and as she does her eyes stay in the same place while her head moves, then it fixes itself. (00:16:20)
Other mistake: The Air People were wiped out by the Fire Nation through Suzon's Comet enhancing them, we even have confirmation of this by the dishevelled exposed skeleton of a VIP of them (presumably any Air Nomad who escaped would have given Monk Giatsu honorable burial), but we never see scorch marks on the Air Nomad buildings or the aforementioned skeleton.
The Ember Island Players - S3-E17
Other mistake: When they are watching the recap play Toph says she can't see it because they are on a balcony, but when her character comes on she describes him as buff.
Chosen answer: Toph had Katara use it to send a letter to her parents. It never returned.