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.
Continuity mistake: The Enterprise looks way more damaged than it did at the end of II. In the Wrath of Khan, Enterprise takes nearly all her damage on her port (left) side. At the start of The Search for Spock, she is damaged about equally on both sides, and still more so. (00:12:05 - 00:13:20)
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. (00:13:30)