Continuity mistake: In the scene when they discover the Wizard behind the curtain there is a change in the equipment on his wall. At first the big lit up gauge thing with the two levers above it is lit, about a minute later when the Scarecrow gets his diploma the angle has changed and the gauge thing is dark.
Continuity mistake: After the talking trees throw their apples at Dorothy she walks up a small hill and finds the Tin Man. If you look behind her there's a house across the Yellow Brick Road. We just saw this exact same spot in the previous shot as they were being pelted with the apples and the house is not there.