Revealing mistake: As the Tin Man busts down the Witch's door with his ax if you look closely you'll see a slight edit in the film right before he takes his last swing. What they did was unlock the door before the last swing so it would open. You can see the dead bolt retracted as the door swings open.
Deliberate mistake: Before Dorothy is sent to Kansas there's a close-up of her shoes. All of the group's feet, Glinda's dress, and the rest of the elements around are gone and replaced by a black background. Despite being a deliberate directorial decision to give the shoes a more dramatic focus, it is really awkward to see everything disappear, nevertheless.
Continuity mistake: When Toto escapes from the witch's castle and runs off to the Tin Man, the Scarecrow and the Lion to lead them to rescue Dorothy, he can be heard barking. As he is running up to the three, he is still barking, but his mouth is quite still and his tongue is hanging out.
Other mistake: The Witch's castle is shown twice, the first time is in the beginning of the scene when Dorothy is held in her room, and the second time is when Toto brings the three back to rescue her. Each time the painting is almost exact, even down to the lights in the windows of the castle. The only thing that changes is the walk way up to the castle, which is entirely different each time.