Revealing mistake: When Britta slides through Jeff's legs to shoot the disco guys, you can see the skateboard-type thing the actress is lying on.
The Science of Illusion - S1-E20
Visible crew/equipment: After Britta accidentally bumps the gurney into the window, she grabs the body bag in a failed attempt to keep the corpse from falling out into the quad. Right as she grabs the body bag, look on the ground behind her. Someone slides a mat in behind her for her to fall onto. (00:13:50)
English as a Second Language - S1-E24
Continuity mistake: When the class leaves the Spanish classroom so Chang can talk to Jeff, in one shot Annie only leaves her audio recorder on her desk, but in the next shot her textbooks are suddenly there.
Spanish 101 - S1-E2
Continuity mistake: When Pierce pours him and Jeff a glass of whiskey, two glasses are shown and he pours whiskey into only one of the glasses. In the next shot, both glasses have whiskey in them. (00:09:25)
Answer: The implication is that Jeff will still be teaching at Greendale after Annie and Abed leave. As for who became vice dean? It's never shown, since it's not really important to the series.
TedStixon