The Northern Air Temple - S1-E17
Continuity mistake: When Aang, Teo and Katara are walking up to the door of the Air Temple Sanctuary, the pipes which make it open are in a horizontal position, but in the next shot, they are vertical. (00:09:20)
Continuity mistake: When Azula says: "What is she doing?" Ty Lee is standing behind her. When the shot cuts and Ty Lee shrugs, they are standing side by side,. (00:20:30)
Continuity mistake: When Azula is asking Zuko whether or not he thinks the avatar may have survived, Zuko goes into a flashback. When he snaps back to real time, Azula has her hands behind her back. When the shot cuts, her hands are instantly by her side.
Continuity mistake: In the shot after Mai says "Saving the jerk who dumped me" you can see the rods which the prison guards used to slow down the mechanism of the gondola are nowhere near the wheel. (00:19:55)
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.