Continuity mistake: In Kirk's log entry, which he somehow makes from jail without a tricorder, he states that 5 witnesses heard him speak to the "spirits." This isn't true, and Kirk ought to know it. The crowd rushed in through an archway after he spoke to Spock and McCoy, and they were all too far away before to hear him. (00:10:50 - 00:26:00)
Plot hole: The law officer who arrests Kirk recalls hearing Kirk call the "spirit" Bones. But he wasn't there when Kirk spoke to McCoy. He rushed into the scene several minutes later. If he's lying, how would he know that Kirk used the name Bones? He wasn't there to hear it. Even if he had been, Kirk did not refer to McCoy as Bones in that conversation. (00:10:50 - 00:19:50)
Continuity mistake: When the prosecutor visits the cell, Kirk's hands on the bars change positions when the camera angle reverses. From the inside, there's one bar between Kirk's hands. From outside, there are two. (00:28:10)
Revealing mistake: Zarabeth tells Spock that Zor Khan only left her what was necessary to survive in the ice age. Mascara, eyeliner, lipstick and hairspray were apparently considered survival necessities. Yes, we know the actress "had" to wear make-up. But in a prehistoric setting, it shouldn't look as though she did. She could and should have been given a more "natural" make-up job - and an unlacquered hairdo. (00:38:25)
Audio problem: In Zarabeth's cave, when McCoy says, "You've been dishonest with me, Spock," his lips are completely out of sync with his words. (00:42:00)
Character mistake: Spock tells Zarabeth he came from a world "millions of light-years away." The galaxy is only about 100,000 light-years across - the area the Federation has explored even smaller.
Other mistake: The time that Kirk winds up in doesn't look anything like the what he was viewing on the disk.
Suggested correction: The entire planet isn't going to look exactly like what he saw on the disk. The disk will show random scenes to give the viewer an idea, but we can't expect the whole planet to match that image.
Visible crew/equipment: The shadow of a boom mic is seen against the rock walls of the cave follows Spock and McCoy when Spock pulls McCoy off Zarabeth.
Other mistake: As Kirk nears the portal, the edge shimmers just before he gets to it. As he goes through it, his actions are greatly slowed rather than natural.
Continuity mistake: After Kirk knocks the jailer out, there's a bit of stew on the cell's door above the lock. After Kirk drags the man into the cell and closes the door, the bar above the lock is clean and dry.