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.
Other mistake: Pay close attention to the taxi cab at the end of each show when it drives away the contestant who was just fired. Look at the taxi number on the passenger side door when it is first opened. You then see the number change after the taxi drives away.