Deliberate mistake: At the end of the film, Gaston shoots the Beast with an arrow and pushes the Beast out the window. When the Beast lands on the edge of the castle rooftop, the arrow has disappeared. Even if part of the arrow has broken off, a piece of it would still be in his back. [This, like many other incidents in the film, is a longtime Disney animated tradition. The company seems to believe that violence is less unpleasant if all evidence of it disappears quickly. This is why so many cuts seem to disappear or become less serious. The arrow vanishing is just another example.]
Deliberate mistake: During the Gaston song in the tavern, Gaston is wearing a pull over shirt with no buttons in front. At one point in the song he rips open his shirt to show off the hair on his chest. The very next second, however, his shirt is back to normal. The shirt is not ripped or disheveled, and it is completely unscathed as if it had never been ripped open or torn apart.