The Enterprise Incident - S3-E2
Audio problem: During the scene where Kirk is talking in the briefing room, the director was obviously not pleased with how James Doohan said his lines. Due to budget constraints, instead of re shooting the scene, they simply replaced Mr Scotts dialogue with another one. You can see it since at one point his lips are completely out of sync, even on the remastered edition. (00:06:40)
The Enterprise Incident - S3-E2
Other mistake: When Chapel comes into Sickbay and looks at the "dead" Captain, the camera goes to close up of Kirk, and he wakes up. After she yells for McCoy, the camera goes back to Kirk, and he goes back to sleep. The motions of him waking are obviously played in a slow reverse instead of him quickly closing his eyes.
The Enterprise Incident - S3-E2
Plot hole: In the conference room, Spock informs Kirk that his theory of a Romulan cloaking device kept the sensors from detecting the ships. However, in S1:E14, "Balance of Terror", they figured out the cloaking device, and how it worked. Unless he was feeding into the deception, Spock shouldn't have had any theory when it was a fact.
The Enterprise Incident - S3-E2
Plot hole: Despite the Romulan ships keeping a close watch on the Enterprise, when they beam the disguised Cpt Kirk over to the Romulan vessel it goes undetected.
The Enterprise Incident - S3-E2
Continuity mistake: When the guard opens the cell barrier, the light bars turn off, then slide back. As they come back out, they're already on, rather than the reverse of the opening procedure.
The Enterprise Incident - S3-E2
Other mistake: As the Romulan commander goes to answer the intercom, the switch she touches to open the link is already on, probably left that way from a previous take.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.