At the end, Charlie shouts "When I came aboard!" This was in regard to the Enterprise liking him initially. Then a couple minutes later, while stating he didn't want to leave with the aliens, you hear him shout again "When I came aboard!" and makes absolutely no sense the second time. An audio overdub in the wrong place.
McCoy refers to Spock as "Doctor Spock" (not Mister). Correction - McCoy says "Probes have been known to be wrong, (pause) Spock" - but Spock talks over him saying "Doctor, are you speaking scientifically...?" It's Spock who says "Doctor", not McCoy.
When the captain of the Antares is trying to warn Kirk of Charlie's abilities, Kirk is in a corridor talking to Charlie about not slapping girls on the butt. He says, "I'm on my way to the bridge now," and gets on the turbolift wearing his usual yellow shirt. When he arrives on the bridge he is wearing the green tunic.
When Kirk and Spock enter Rand's quarters, Charlie hurls them back against the wall. When Spock slumps to the floor, we can see that the set wall behind Spock's head has been damaged. However, in later shots in the same scene, the damage is gone.
When Charlie arrives on the Enterprise, Kirk instructs Yeoman Rand to show him to his quarters. Then later, he is in Sickbay and McCoy says that he will show Charlie to his quarters, which Rand should have already done.
At the conclusion of Charlie's physical exam, Dr. McCoy tells Charlie that he'll show him to his quarters and they start down the hallway together. However in the very next shot, Charlie is alone. He stops to watch crewmen work and even climbs into a Jefferies Tube (he is shown backing out of the tube ahead of two crewmen). Still Dr. McCoy is nowhere around.
When Charlie walks on the bridge after taking control (when Spock is forced to speak the lines of various literary works), the pattern screen on Spock's console is still when Kirk and Spock are shown at an angle from the front of the bridge. In the head-on shots of Kirk and Spock, the screen shows the moving pattern.
Shortly after the scene where Kirk tells the cook that he wants the synthetic meatloaf to look like turkey, he and Charlie get on the turbo lift on their way to the bridge. When Kirk enters the Turbo lift, he is wearing his regular gold captains uniform. However, when he exits the turbo lift seconds later, he is wearing his alternate green-tinted uniform with the rank insignia on the shoulders.
If Charlie was capable of making all the phasers on the ship disappear, why would Kirk believe that Charlie's quarters would be able to hold him? Because Charlie only attacked in reaction to what he construed as provocations from other people. Isolating him from other people would prevent such provocations. Afterwards, Kirk tried to trick Charlie by sending him to a force-field protected brig instead of his quarters. While that didn't work, it was a nice try in a desperate situation.
When Kirk, Spock, and McCoy are discussing the possibility of intelligent life on Thasus helping Charlie to survive after his ship's food supply had been exhausted, even without Spock calculating the odds (which he had not yet begun to do in the series), McCoy should know how improbable it would be that Charlie could survive on native flora for thirteen years without making a single mistake and ingesting something poisonous. Unless, of course, the planet didn't have any poisonous plants or dangerous animals - a possibility McCoy considered more likely than having a mysterious, undetected civilization inhabiting the planet.
When the Thasian returns everything and everyone to the way they were before Charlie made them vanish, Yoeman Rand reappears on the bridge in her nightgown. But she had vanished while in her quarters. Crewman Sam, who had vanished from the gymnasium, didn't reappear on the bridge.