After Garth reveals his deception to Kirk and changes back to his own form, he releases the rest of the patients from their cells. As Marta comes out of her cell she removes her outer dress and lets it drop to the floor where it disappears. In subsequent shots the dress is gone.
During the seduction scene between Marta and Kirk, Kirk is alternately lying down or sitting halfway up.
Even if Garth-as-Kirk managed to take control of the Enterprise, the crew would have known something was very wrong as soon as he gave orders to attack Antos IV. Garth overlooked this because he is a madman. Per the "supervillain's master plan with a fatal flaw" trope, a mad genius must lack a crucial element of common sense, so that the heroes can win.
When the Enterprise fires phasers at the planet, the beams diverge from the ship, but then converge toward the target on the planet's surface. This was a common error when showing the ship's phasers firing.
When Garth (as Kirk) fails to get on board the Enterprise due to not knowing Kirk's pre-arranged passcode, he loses his temper and pounds the console in the room. Although William Shatner tries to go easy on the prop, his pounding still pops the left knob loose.
The Andorian reacts to the explosion before it happens.
Throughout most of the dinner scene, when Marta and Garth are bickering back and forth, his henchmen break character and just barely resist laughing. It could be possible it was scripted that they appear amused, but it's more likely that it's the actors, not the characters trying to restrain laughter.
William Shatner's stunt double is plainly a couple of inches taller than him. This is evident in the scene where the two Kirk's fight each other.
Several scenes on the Enterprise bridge show panels in the background that should be lit up, yet the displays are dark.
As Spock approaches the control room, Garth sees him on the monitor and says, "This could be amusing", but it's clearly Leonard Nimoy's voice.
When Governor Cory is first being held by Garth, he is supposedly suspended in mid-air, presumably by a force field of some type. But in at least one shot, the angled brace that's holding him up can be seen behind him.
Governor Cory explains to Kirk that Garth can change his appearance at will due to his control of his body cells, but that does not explain how his clothing changes as well (a typical hitch with sci-fi shape-shifters).
It is doubtful that a mental health facility would have the necessary equipment and materials to develop an explosive.
None of the staff members who work at the asylum, with the exception of Governor Cory, are ever seen. Only one is even referenced, when Cory states that it was a guard that released Garth from his cell, thinking he was the governor. There is no mention of what happened to the staff.
Even if Garth can assume the appearance of others by changing his body cell structure, it doesn't explain how he can perfectly mimic the individual's voice as well.
Scotty has ample time and opportunity to beam down a security team when Garth, in Spock's form, deactivates the force field. The security team is already in the transporter room and several minutes pass before Kirk forces Garth-Spock to give Scotty the code response, leading to Garth reactivating the force field several minutes later.
Garth, as Kirk, is having a tantrum because Scotty will not let him beam up to the Enterprise. During his transformation back to Garth, while kneeling and banging his hands on the floor, a piece of Garth's uniform can be seen falling off and spinning on the floor near his right hand.
Initially, Garth's explosive invention is his latest (according to Gov Corey). Later, the 'chair' is his latest invention. Later, the explosive is once again, said to be his latest invention.
Kirk makes an incorrect statement. In chess, there has to be a finite number of counter moves.
Whenever he is addressed by his proper title "Captain", Garth immediately and angrily corrects the speaker, insisting that he is to be called "Lord Garth", in all but one instance. To maintain the character's mania, Garth should have reacted similarly when Kirk says, "I am primarily an explorer now, Captain Garth," but he does not.