Continuity mistake: In the scene where Bilbo, now invisible, sees Gandalf and the dwarfs run past the cave and he can't join them because Gollum blocks the way. When Bilbo finally runs past Gollum and kicks him in the face and knocks him down, the very next shot shows Gollum's body lying in the opposite direction, away from the direction Bilbo kicked him.
Continuity mistake: As Thorin is picked up by the eagle, the eagle's talons roll him towards the camera. Both of the eagle's feet are in view and you can see all the way down Thorin's body to his boots. Orcrist is nowhere on his body during this initial shot of the pickup. As soon as the shot changes (where the Oakenshield drops free) and both of the eagle's feet come back into view, Orcrist is now on Thorin's body, running over his stomach and left leg. Had it been there before, it would have been visible when the eagle rolled him towards the camera.
Continuity mistake: In the scene where Bilbo spares Gollum's life, Gollum turns around facing the invisible Bilbo and extends his right hand out to a rock. The far shots from in front of Gollum show his arm almost fully stretched out and his hand away. But the shots from behind Gollum show his right arm bent on the rock with his hand much closer and level with his head.
Continuity mistake: Shortly after Bilbo finds the Elvish sword that turns blue in the presence of Orcs and Goblins there's a fight with Orcs at the entrance to the hidden valley. However, the sword doesn't glow blue. In subsequent fights with Goblins and Orcs the sword does glow blue, even when sheathed.
Continuity mistake: At Bag End, Bilbo recovering from fainting, sits in an armchair. Gandalf is speaking with him. Behind Gandalf is an alcove with an unlit candle, which is lit in the next shot, although not illuminating any of the alcove, then it's unlit again.
Continuity mistake: During the song "Blunt the Knives", Balin is seen in a shot from behind sitting at the end of the dining table. He is shortly afterwards seen dancing with a broom two doorways behind. The next shot then shows him sitting down again, bouncing plates off one he is holding.
Other mistake: Just after he arrives at the gathering Thorin states that he wouldn't have found Bag End had it not been for the mark Gandalf made on the door. Bilbo then claims that 'There is no mark on that door, it was painted a week ago'. However in the earlier scene where Gandalf makes the mark the paint work is cracked and weathered, more than it would have done after only a week.
Other mistake: When an invisible Bilbo is about to jump over Gollum to exit the cave, you can see Gollum prepare himself for impact just before Bilbo jumps over him. This isn't logical if Bilbo is invisible.