Other mistake: In the Briefing Room, while Kirk is feeling the effects of the disease, Spock tells Scotty that he will call the restart formula in from the bridge. But just a moment later, Kirk enters the Bridge, and Spock is in the Engine Room working on the settings and formula with Scotty.
Visible crew/equipment: After McCoy examines Charlie and they talk about how Charlie learned to talk by just listening to the ship's tapes, when Charlie stands up the two actors' marks can be seen on the floor - the short one for Charlie and the longer one for McCoy, where they both will stand momentarily. (00:04:56)
Plot hole: Pike says he doesn't want women on his bridge, Number One being the one and only exception, and he keeps ordering Colt off it. His apology to Number One indicates that A) he doesn't think of her as female, and B) she's the only exception to his rule. So how does he overlook the very female crew-woman seated at the science station? (00:06:55)
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different. (00:40:50)
Plot hole: Because of the magnetic ore on the work clothes of the man who beamed up, the transporter made duplicates of everything put through it, and so couldn't be used. This was further complicated by the control and power circuits being blasted by the "evil" Kirk. However, the ship carries a number of shuttle craft, which no-one mentions.
Revealing mistake: When the laser cannon starts firing at the Talosian rockface, a styrofoam block falls and rolls away on the left side of the screen. After a brief shot of the crew, we see the cannon still firing, and the exact same block falls and rolls off screen on the left again. (00:30:15)
Other mistake: Spock and McCoy have a brief argument over whether Charlie could have survived on his own all those years. Because they're talking at the same time, Spock saying "Doctor" as McCoy says "Mr. Spock," the closed captioner apparently misunderstood McCoy's line. The caption has him calling Spock "Dr. Spock." (00:09:15)