The first woman and man who enter the disintegration machine and die can later be seen standing in the corridor near it as Kirk and Spock blast it with disruptors.
The Eminians attack the Enterprise with a sonic weapon. However, sonics would not be effective against an orbiting starship, as the vacuum of space (or even a planet's upper atmosphere) would not conduct sonic energy.
When the Enterprise is attacked with a sonic beam (assuming a sonic beam could even travel through a vacuum), the decibel reading is 18 to the 12th power. This would be about 1.15 quadrillion decibels, and since the decibel energy scale is exponential (every 10 decibels is an increase in power by 10x, so an increase of 30 decibels is 1000x), 1.15 quadrillion decibels would be more energy than exists in the entire universe.
Spock distracts a guard by telling him there is a multi-legged creature on his shoulder, an obvious lie. There is a common misconception that Vulcans are incapable of lying, which is absurd. If they could not lie, they could never be sent on undercover work, which we see Spock and other Vulcans doing on numerous occasions.
When Kirk fights with the guards outside Anan's cabin, the sheet rock wall of the set is damaged in several places during the scuffle.
When Kirk orders the 'General Order 24', as the guards are trying to restrain him, his hand bumps the central box on the desk with the two mini-view-screens. The bump moves it - indicating the equipment on the table is merely resting on it, and not built-in, as would be expected.
In the opening scene, just after Ambassador Fox tells Captain Kirk to obey his order, he steps up to the upper level of the bridge, slightly tilting/dislodging the captain's chair. Causing it and Kirk to wiggle. Surely the structural integrity of a 23rd Century star ship bridge deck floor would not allow this to happen.
After Kirk destroys the war room computers and the scene shifts to the corridor, in the ensuing dialogue, Kirk and Spock refer to the Eminiars' foe as the Vendikans instead of the Vendikars.
In the beginning of the episode when Kirk orders a Yellow Alert , the Red Alert alarm sound is heard.
In the end credits, George Rutter is listed as "Scpipt Supervisor".
When Kirk and the landing party first visit Anan7, you can clearly see the tape marks on floor identifying where each crew member is supposed to stand after they walk into the scene.
During the beginning scene, Kirk mentions they are arriving at star cluster NGC 321. NGC 321 is not a star cluster but an elliptical galaxy.
The Enterprise's shield are up and Scotty refuses to lower them because he fears (correctly) an attack. Ambassador Fox insists that he be beamed down, and again Scotty refuses to lower the shields in order to do so. However, in the next scene, Ambassador Fox has somehow beamed to the surface (at 34:05), and Scotty doesn't lower the shields until later, after he talks to Kirk at 47:51.
Anan 7, when summarizing the effects of the landing party's actions, states that one of the disintegration chambers has been destroyed and that they have already fallen behind in their quota. If one person was disintegrated per minute, one chamber could only handle 1440 people in 24 hours. The casualties from the battle were in the millions. Hence, any falling behind could hardly be attributed to the destruction of the landing party's actions, which would also probably not affect things outside the building.
When Ambassador Fox's aide is struck by a disruptor beam (How, since he was behind everyone else?), the Ambassador just checks on him and goes on - no report to anyone that he was hit. There is no indication of the aide's state - alive or dead - he just seems to disappear for the remainder of the story.
The Enterprise initially enters orbit because the ambassador disregards a 3-digit code intercepted outside communications range. But the planets have the technology for a sub-space data link between computers several planets apart, so they could certainly have explained the situation in more detail before the Enterprise entered the war zone.
Ambassador Fox insists on making contact with Eminiar 7, even after the government there tells the Enterprise to leave. This is a clear violation of General Order One (commonly called the Prime Directive), and, while the Federation diplomatic corps is a civilian agency and not part of Starfleet, the Federation civilian government presumably holds by the same philosophies as its military.