Revealing mistake: The graphic display showing the Enterprise lowering shields is actually just a repeat of the earlier display showing the vessel raising her shields...played backwards. The giveaway is the screen still reads "Activating Deflector Shields" even though Scotty is supposed to be dropping them.
Revealing mistake: When Sybok lifts Kirk off the ground in the shuttle bay, wires are visible holding up Kirk by his legs.
Revealing mistake: In the beginning of the movie where Kirk falls off the mountain, it's very obvious that he's falling against a backdrop.
Revealing mistake: On Sha-Ka-Ri, after Kirk says "I seek proof", and the camera goes to "God", then back, you can plainly tell it's a stunt double, and see the wire connected to the harness he's wearing to pull him backwards after getting zapped.