When The Blue Spirit frees Aang from his wrist irons, he fails to free Aang from his leg irons, and turns away, stowing his swords. However, Aang is running with him in the next scene.
During the scene where Aang is being tested by Zuko's uncle Iroh, Iroh pours a cup of water onto the table and it coalesces into a circle (at around 26 mins). In the next shot, Iroh places a rock on the table where the water had just been (at around 29 mins), but the water has inexplicably disappeared.
In the scene with Sokka and Katara, just before Sokka is breaking the ice. You can clearly see a large ice formation just behind and the the left of them. During the ice breaking scene it is still there, yet when the ice is cracking and they back the way they came its gone. Also, in the over head shot the terrain all looks flat.
During a large battle scene between the Fire Nation and the Northern Water Tribe, the camera pans to reveal a Fire Nation soldier fighting with no one.
During the first conversation between Fire Lord Ozai and General Zhao we can see that Ozai's eyes move from left to right while he looks in the direction of Zhao's chest, thus revealing the use of a prompter.
When the flying lemur jumps on Aang's shoulder, Aang's hands supposedly touch and pet the Lemur. But it is clear that the lemur is computer generated as Aang's hands do not actually touch the creature, they seem to hover about an inch or so above him.
When Aang first starts airbending at the prison camp, the gusts of air that he's generating don't actually ruffle anybody's clothes or hair or even the dirt and sand beneath their feet, revealing that it's simply a special effect.
When admiral Zhao is killed, the snow on the ground dissolves in a soapy, white mix in the water poodle around his body, unlike real snow.
When Aang and his friends are arrested in place where is occupied by firebenders, they are put into a prison for earthbenders. However, the entire prison is made out of earth, as Aang points out, to which they should have known this already, defend themselves and escape.
The Old Man (Randall Duk Kim) was conspiring against Aang to turn him in to the fire nation soldiers for a reward when he arrived at the air temple. How did the Old Man at the Temple and the soldiers know he would be there? There's no scene where the Old Man goes to inform the soldiers that the Avatar has arrived.
When Zuko takes Aang onto his ship, Iroh puts a rock, lit candle and water in front of him, to which all three pull towards him, showing he is in fact the Avatar. However, Aang walks by several rocks, water and fire and they never react to him, without bending.
Katara could have easily ended the fight between herself and Zuko in the Moon Spirit temple if she had simply doused his fire source with water, since firebenders cannot create their own fire.
When Katara first sees Appa rising in the air, she describes it as floating. Appa is actually levitating, not floating, as objects only float when they are positively buoyant, or less dense than their surroundings.