Other mistake: When Savic is sleeping, a burst of energy from planet pushes the tree she is leaning on upward. There is a Klingon above that almost gets hit with it. Savic should have seen him and started moving with Spock. This might have been a scene meant to be used later at the climax. (00:57:00)
Revealing mistake: When Kirk and Kruge are fighting, a piece of the cliff breaks away and wiggles down the side of the cliff rather than falling like a rock. (01:22:55)
More quotes from Star Trek III: The Search for Spock
Trivia: At the beginning of the movie Kirk rescues McCoy from a jail of some sort. Before leaving Kirk asks McCoy how many fingers he's holding up and does the Vulcan hand thing. However, Shatner had severe difficulties putting his fingers into place, they just wouldn't hold into position. So the crew wound up wrapping fishing line around Shatner's fingers, he would put them into position out of the camera and the shot quickly jumped while his fingers where still in place. If you look closely you see the line around his fingers. (Note that this problem occurred several times during Star Trek, actually, Nimoy and Leonard seem to be the only people doing this without a problem.)
More questions & answers from Star Trek III: The Search for Spock
Chosen answer: Kruge only had the proposal video that talked about what Genesis was. He didn't have any actual data on how Genesis works or how it could be created. Having footage of a nuclear blast doesn't give you knowledge about how to build a nuclear device.