Deliberate mistake: Before each season begins filming, each potential apprentice is filmed leaving Trump Towers and entering the taxi as if they were just fired. Once the person is actually fired on the show, this becomes an obvious mistake as hairstyles, etc. change from the boardroom scene to the cast member's exit.
Continuity mistake: In the episode where they overhaul a 1976 blue van, there is a shot of the white donor van being carted away. Yet later on in the same episode, the white donor van is back in the workshop.