Continuity mistake: When Gaston and Le Fou approach the village on horseback, they watch Belle through a telescope. In the shot from behind, Le Fou has to stretch out to reach the telescope when Gaston passes it across and his fingers barely touch the end of it, but when the view changes to the front the horses are closer together and he is holding the telescope firmly in the middle.
Revealing mistake: When Belle exits her house, the reflection on the window by the door shows a rectangular flat roofed building, nothing to do with the old gable roof we see in further shoots. In the making-of we are revealed that the prop building is in fact rectangular and flat, with a green screen on top to allow for the CGI.
Continuity mistake: In the opening number, Belle picks up a loaf of bread at the start and puts it in a bag around her waist. Later in the number it's not there. There's a deleted scene which explains the disappearance, but with that scene missing, the bread just disappears for no reason, hence it's a mistake.