At one point during an argument with Spock and McCoy, Kirk complains about them involving themselves in his personal affairs. However, the "affair" ceased to be merely personal as soon as he involved the ship and crew, and as an experienced officer he would have been well aware of it.
While McCoy is enjoying a "drop of the true" in sick bay, he offers Spock a drink. Spock explains, "my father's race was spared the dubious benefits of alcohol." To which McCoy sardonically responds, "Oh, now I know why they were conquered." However, in Star Trek: The Immunity Syndrome Spock explains to Kirk that, "Vulcan has not been conquered within its collective memory. The memory goes back so far that no Vulcan can conceive of a conqueror."
Mr. Spock reports that Governor Kodos had his 4000 victims killed efficiently and "without pain." The brutally half-destroyed face of Dr. Leighton, also attributed to Kodos, seems to be at odds with this claim.
In the cocktail party scene Captain Kirk sips from his drink. Then soon after Lenore enters-whom Kirk has never met before-after a brief exchange asking if the drink he's holding is for herself. Kirk displays his lack of manners saying "Why not?", giving it to her knowing he just sipped from it.
Riley wears a gold uniform shirt indicating he's a member of the Command Division. However, he is seen serving both the Communications and Engineering Departments, and should be wearing a red Ship's Services Division shirt.
While Star Trek anticipated a number of technological and scientific innovations, it apparently failed to predict DNA testing, which could have resolved the question if the body burned beyond recognition found on Tarsus IV was indeed that of Kodos the Executioner. The prequel series Star Trek: Enterprise does make frequent references to DNA identification.
The drama hinges on the idea that if "the last few eyewitnesses" die, then the knowledge of what Kodos looked like will be lost. But Kirk easily orders up Kodos' mug shots and voice prints to compare with Anton Karidian.
A phaser set to overload begins making noise as soon as the setting is engaged. However, the phaser hidden in Kirk's quarters doesn't start making noise until long after it was placed, in the middle of Kirk's conversation with Spock.
When frantically searching for the overloading phaser in his quarters, Kirk throws a lot of books to the ground but completely ignores a number of different containers and boxes that were more likely to have a phaser hidden inside them.
Spock discovers the Tarsus IV connection by asking to computer to correlate the historical data on Kirk, Leighton, Riley and Karidian and identify any common episode or experience. The computer reports in the affirmative. But it's previously clear that Kirk isn't certain about Karidian being Kodos at least in part because there is no record placing Karidian at Tarsus IV.
The goof items below may give away important plot points.
Incorrectly regarded as goofs
A Phaser set on kill will disintegrate the person its beam hits, yet Karidian falls to the floor intact. However, there does seem to be a middle ground setting, which is shown in other Trek entries as well.
Kirk quarters the actors on his ship, believing the murderer of Dr. Leighton to be among them, with Anton Karidian as the prime suspect. Yet the actors, including Karidian, are apparently allowed to walk about the ship freely, and Lenore is able to walk undetected into sensitive areas and make attempts on the lives of Kevin Riley and Captain James Kirk. Karidian cannot account for his whereabouts during the attempts. If Kirk had assigned surveillance to him when he came on board (as common sense dictates), he would have been cleared of involvement in the attempts.
Captain Kirk orders Anton Karidian to read a text into a wall communicator in order to determine whether the actor is in fact Kodos the Executioner. Kirk tells Karidian that electronic voice matching will prove infallibly whether or not Karidian is Kodos. But when the test results come back - Mr. Spock calls them a match - Kirk insists that the "infallible" comparison still isn't close enough.