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: Van on the Blue Team is asked to leave Hell's Kitchen by Chef Ramsey and is shown walking out. Soon after, when the Red team is nominating team members for elimination, the camera shows Nick on the Blue team and Van's Head is in the shot. If he was just eliminated how could he still be in kitchen. (00:57:00 - 00:58:00)
Other mistake: The Zeds infect others through directly touching them. This rule extends to touching their clothes and backpacks, which makes no sense.