Continuity mistake: When Jeff goes into Ian's office and sits down, he puts his book down so the spine is facing the viewer's right and puts his drink on top of it. Later, without touching the book, it is rotated 90°.
Continuity mistake: The study-room on the second floor of the library, right next to a set of stairs, and the area isn't particularly bright. In the rest of the series, the study room (which is meant to be the same study-room) is now located on the first floor of the library, nowhere near any visible stairs and the area is much brighter and more well-lit. As often happens, the pilot was filmed before any other episode, and the setting was changed in the meantime.
Continuity mistake: In the very first shot, we can see there's a man with a dark coat, sunglasses, short hair and earbuds standing directly in front of the dean. When it cuts to the opposite angle, he's no longer there. When it cuts back to the first angle a few shots later, he's suddenly back.
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