Continuity mistake: When Dorothy is at the end of the movie confronting the Wizard about going home, since she killed the Witch, Toto goes over to the curtain and you can see him tied up to it as he opens it. Dorothy then goes over there and automatically has him in her arms - she didn't even need to bend down and pick Toto up.
Continuity mistake: When the Tin Man, Scarecrow, Cowardly Lion and Dorothy are cornered in the witch's castle, the scarecrow grabs the Tin Man's axe and chops a rope, releasing the chandelier, which falls on the guards. In the next shot, however, as the witch is shrieking "seize them", the guards give chase with only 1 or 2 laying on the floor.
Continuity mistake: After the Witch throws a fireball at the Scarecrow in the woods and disappears, Dorothy is holding Toto. But when it cuts to a closeup of her, it looks like her shoulders are extended. When it cuts back to the wide shot, she indeed has her arms around the Scarecrow and Tin Man, even though she never put Toto down.
Revealing mistake: In the scene when the Wicked Witch of the West bends down to retrieve the ruby shoes from the feet of her sister sticking out underneath Dorothy's house, there is one quick shot in the middle of the scene that shows her in front of a screen with a picture (film) of the house on it. Maybe that shot was filmed after the original set was unavailable? That would seem to be the only possible and reasonable explanation why such a shot would be necessary.
Continuity mistake: When the Lion is singing about being King of the Forest, the TinMan breaks a flower pot to "crown" him. Pay close attention, as the break marks on the pot change. In one shot there's a big jagged edge sticking up (obviously longer than the others) and in the next scene most of the edges are even.