When Kirk and Spock are discussing the identity of the penal colony escapee (Dr. van Gelder), the camera angle goes back and forth between them. When looking at Kirk, the background wall (correctly) curves to the right, past communications and the turbolift. But the reverse angle at Spock *also* curves to the right; however, it should curve to the left, towards the main viewscreen.
After Kirk orders "security alert 3" and the alarm sounds with its flashing red light [3 times, behind Dr. van Gelder at a corridor intersection], though the alarm continues to sound there are no more flashing lights, either in the corridors or on the bridge, where there are neither (alarm sounds or lights) right up to when Kirk orders "Reverse course!" after he and Spock neutralize the fugitive.
When Dr. van Gelder enters the bridge and demands asylum, the close up shots use red lighting on the actors. In the wide shots, the lighting is normal.
Among its many depictions, the bulkhead above the transporter room doors does not have a centered red alert light.
Near the end when Kirk bursts into the treatment room he turns the power down with his right hand, in the next shot the same hand is holding his phaser.
When Uhura reports that the transporter officer is found unconscious, she says that the box was found with the lid open. However, when the assailant leaves the box and the transporter room, the box is clearly closed.
Transporter Operator Berkley is credited as an ensign, but the uniform is clearly that of a lieutenant.
Around the 46 min. mark, where Dr. Adams has been left in the treatment room chamber alone, being exposed to the beam on full force, there is a shot of the control panel and the readout says "OFF".
When Dr. van Gelder is on the loose on a ship, a security guard is deployed to the bridge. He stands a few feet away from the turbolift doors, facing the interior of the bridge (which makes it nicely convenient for Dr. van Gelder to hit him from behind). He should have been standing further away from the turbolift (possibly by the railing) and facing towards it; was he expecting Dr. van Gelder to gain entrance to the bridge via the viewscreen? He also never moved when the turbolift doors opened; he should have instinctively turned around, especially since he was supposed to be guarding the bridge from the intruder.
Captain Kirk and Spock act surprised to see that Doctor Noel is a woman. It is highly unlikely that either the captain or first officer would not be aware that there is a female doctor on the crew of their own ship, even if they did not remember her name.
When Kirk was in the neural neutralizer chair. Dr. Adams takes the controls away from Helen Noel. One of Dr. Adams' orders to Kirk is to drop his (Kirk's) phaser, which he does. Then, Kirk is ordered to drop his communicator but he musters the effort to try to contact the Enterprise (which would be futile due to security measures). If Kirk had the energy to call the ship, then he could have used the phaser to shoot the beam emitter for the neutralizer thus breaking its effects on him.
When Dr. van Gelder reaches the Enterprise's bridge, he attacks and disables a security guard. McCoy is on the bridge at the time, but even after Dr. van Gelder is subdued, he never so much as glances in the unconscious guard's direction.
After Dr. van Gelder is seen assaulting and injuring (perhaps even killing) several Redshirts, Kirk tells Dr. Adams that there were "no casualties" to report on the Enterprise.
Mr. Leslie, manning the bridge Engineering station, is wearing the plain red shirt of an ensign/yeoman instead of the single gold braided ribbon sleeves of a lieutenant.
After Kirk leaves the transporter room and a chastened Mr. Berkeley goes to get some equipment to move the beamed-up box and refresh his knowledge of penal colony transport procedures, his assistant is left alone to be attacked by Dr. van Gelder - without any sign that Berkeley had left the room.