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.
Character mistake: Sybok tells Kirk that no one believed the world was round until Columbus proved it. Someone who studied at the Vulcan Academy should know that the Greek philosophers had proved the world was round and calculated its circumference centuries earlier. Columbus never set out to prove the world was round, just to find a shortcut to India.
Visible crew/equipment: When the being they think is "god" shoots Kirk with lighting eyes, you can see the pull wire on his back as he flies back.
Continuity mistake: At the camp fire the whiskey bottle never empties. Almost empty, then full again.
Factual error: When Kirk falls off El Capitan, he is shown falling all the way past the tops of the trees. I have been to Yosemite and the trees are tall, but not tall enough to give Spock enough time to slow then stop Kirk's fall.
By the time Kirk reaches the top of the trees, he is traveling at terminal velocity (assuming Kirk was halfway up, about 2000 feet from the ground, and an initial fall rate of 32 feet per second, per second), approximately 120 mph, about 200 feet per second. If the trees were 1000 feet tall and Spock caught Kirk right at the top of the trees, that would give him 5 seconds to slow down and stop Kirk. Impossible. Fortunately, in the movie he was able to accomplish this feat, or it would have been a very short movie. :).