Continuity mistake: Early in the film when a lifeform was detected in Mr Spock's quarters, you see a blip on the screen inside the old Enterprise, not the new Enterprise, which was almost a complete rebuild, with a very different layout. Notice the outline of the ship - different engines.
Continuity mistake: When Kirk, McCoy and Sulu beam aboard the Enterprise we switch to an outside shot and the bridge lights come on, a very bright white light that lights up all three windows, but when we go inside the bridge is very dark with only a few lights on.The main lights come on when they man their stations.
Continuity mistake: When Savvik and David first encounter Spock's photon tube "casket", they open it to reveal the burial robe rather nicely folded with the Vulcan lettering shown. When David reaches in to retrieve it, the robe is very disheveled, lying more lengthwise, and the lettering not so clear.