Continuity mistake: When Maurice is tossed out of the tavern we get a nice view of the front with its large hanging sign, but after Gaston whispers his plan to LeFou once again we see the front, which is quite different. The sign is gone, and now there are horns hanging on the awning, etc.
Visible crew/equipment: When they are moving the catapult, if you look above Robin and Azeem, you can see a crew member in the window above them. He is wearing all black, but you can tell he is trying to get out of the shot.
Revealing mistake: Near the start of the film, when Lt. Harley is watching his love interest show off by doing gymnastics on the tree limbs, several times you can see the actual pole she is swinging on, not the tree limb. (00:08:50)
Visible crew/equipment: In the scene where Jack approaches Parry's bed, a crew member is seen on the right hand side of the shot wearing a walkie talkie. He inches out of frame in 3 or four seconds. (01:51:05)