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)
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.
Plot hole: Spock has absolutely no way to know, yet, that the horta only secretes her corrosive substance when tunneling: he hasn't had time to examine her or to do more than determine that she does indeed secrete a substance that cuts the tunnels. So he should at the very least scan the piece of her that falls off before he picks it up with his bare hands. Major lapse of logic, which given his character goes beyond a character mistake.
Plot hole: When M-5 destroys the ore freighter, Dr. Daystrom says, "Fortunately, it was only a robot ship." But Daystrom wasn't on the bridge yet when Spock announced that fact. He didn't have any way of knowing that the freighter was unmanned. (00:23:00 - 00:25:05)
Plot hole: The ambassador and his aide beam down to the planet. This would not be possible because earlier on Scotty refused to lower the screens/shields until the Captain told him to do so. It had already been established in an earlier episode titled "Arena" that transporters don't work with screens up. (00:32:40)
Plot hole: Scotty is receiving images of the launch, supposedly from an orbiting satellite. But many of the camera views are obviously from ground level, and couldn't possibly have been taken from orbit. It's not a rebroadcast from a ground camera, either. The implication is clearly that Scotty is is picking up a live satellite feed. (00:34:30)
Plot hole: Kirk knows that Lazarus is insane and that he wants the Enterprise dilithium crystals. Yet he's not restrained in sickbay and is, in fact, given free run of the ship so that he can knock out the crew in engineering and steal the crystals. Other than to further a woefully weak plotline, this makes no sense whatsoever.
Plot hole: Apparently there is some confusion over the distance between Earth, Starbase 11, and Talos IV. When Spock first meets Pike on Starbase 11 he tells Pike Talos IV is only six days away. Yet when Pike (in the recording) speaks to the Talosians for the first time, he says he is from a star system on the other side of the galaxy. If Talos IV was on one side of the galaxy and Earth was on the other side, it would take hundreds of years at maximum warp to travel from one planet to the other.
Plot hole: Spock claims that history is unchanged at the end of the episode according to the library tapes - how would he know? If history changed, the tapes would change too. And unlike in "City on the Edge of Forever", there's no Guardian around to keep people from being influenced by an altered timeline.
Plot hole: In the conference room, Spock informs Kirk that his theory of a Romulan cloaking device kept the sensors from detecting the ships. However, in S1:E14, "Balance of Terror", they figured out the cloaking device, and how it worked. Unless he was feeding into the deception, Spock shouldn't have had any theory when it was a fact.
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)
Plot hole: It is stated during the episode that Khan was, at one time, the ruler of over 1/4th the Earth's population, during a very key moment in the planet's history (The "Eugenics" Wars). Such a personage undoubtedly would be very well known to 23rd century Earth people, at a level of infamy approximating Julius Caeser or Adolf Hitler. Yet it is only 2/3rds the way through the episode, thanks to a computer search by Spock, that the crew divines his identity. Lt. McGivers at the very least should have almost instantly recognized him.
Plot hole: When Captain Kirk is the last person unaffected by the spores, he's on the bridge lamenting the fact that he is trapped in orbit above the planet since he can't pilot the ship alone. He also can't call for help because Lt. Uhura sabotaged long range communications. but Captain Kirk has somehow forgotten about the shuttle craft. He could have easily escaped on one of those.
Plot hole: When the Medusan Ambassador Kollos and Dr. Jones materialize, the latter is wearing a protective visor and Mr. Spock can see that. Spock then mistakenly addresses Dr. Jones as Ambassador Kollos. Spock's suddenly illogical mistake seems odd: why should the Ambassador need to wear a visor? Shortly after, why do both Jones and Spock wear visors while carrying the Ambassador's still closed gasket to his cabin? (00:01:45)