Continuity mistake: As Max is carrying a birthday cake out of the kitchen, the candle blows out as he walks. But the candle is suddenly lit again when the camera angle changes, as he walks into the living room.
Continuity mistake: As the ecology group's party is winding down, Betty and Leo walk off together and end up by the garden well. During this scene (including while Leo speaks with Betty, The Escort, and Mr. Jordan) the well is visible in almost every shot and changes from being in full sun to full and partial shade several times.
Continuity mistake: Joe takes over Leo Farnsworth's body while Leo is bathing, so it makes sense for him to put on a robe and get dressed after getting out of the tub. Watch, though, as he walks over to the butler who is holding out the boxer shorts - you can see Leo is wearing the same tennis shoes Joe was wearing in previous shots. (I guess Warren Beatty didn't have time to remove the shoes between takes?)