Factual errors: The reason that the Beast needed to rescue Belle was that she was being pulled underwater by a Christmas tree. However, trees float, and Belle would not be pulled down by one.
Continuity: When the Beast enters Belle's cell, he leaves the door open. However, when Forte plays his tunes to bring the castle down, the door is closed and locked.
Factual errors: During the song "Stories" Belle's book is stitched together and we see her holding it in her hands. Viewing it from the back, the binding is in her right hand; unless it's upside down (which it's not, because we can read the writing on the back cover) the spine of the book should be in her left hand.
Continuity: During the course of the movie, the character, Fife clearly has 4 keys/arms, but at the very end of the film, he is seen to clearly to have 6 keys/arms.