Continuity mistake: There are two actresses playing Dorothy (and two dogs playing Toto) as she opens the door to Oz. The first is a stand-in wearing a dull colored dress in order to make it look black and white. After she opens the door and moves out of the way, Judy Garland is who we actually see walking through. Look at the different pleats on the two dresses.
Visible crew/equipment: When Dorothy lands in Oz and exits the house to Munchkinland, the camera pans around the land. When it slowly pans over a large bed of tall flowers, you can clearly see a very large shadow passing over them. Obviously the shadow of camera equipment or a camera crane.
Continuity mistake: When Dorothy walks on the pig sty Zeke walks to his right, looks into, than passes the bucket on the right because it's empty. He then picks up the bucket behind it on the bench and proceeds to feed the pigs. He empties the first bucket then picks up the second empty bucket, which is now three quarters full.
Continuity mistake: When the Lion, the Scarecrow, and the Tin Man are chopping down the witch's door to free Dorothy before the hourglass runs out, they are dressed in the guards' uniforms. We see the axe hit the door, and in the next wide shot they are not in the uniforms. There is a slight view of the uniforms in a pile on the floor, as if the trio had changed out of them between axe blows.