Factual error: If this planet is a duplicate, an exact copy, of Earth, then Lake Okeechobee, the largest lake in Florida, is wrong. Seen from space, it is much larger and more distinct than the lake shown on this duplicate.
Other mistake: In the opening, the planet looks to be turning the same as Earth does. On the viewer it is turning correctly. However, it is too quickly to be accurate, assuming it is a copy of Earth. If so, at this higher speed, the days would be shorter. Also, the ship should be moving at such a speed as to see the planet moving in the opposite direction.
Visible crew/equipment: When Dr. Van Gelder tells Kirk that he wants asylum and his word that he won't be taken back to Tantalus, in the wideshot from behind Kirk as he moves to his right the two tape marks can be seen on the floor beside Van Gelder's feet.
Plot hole: For a penal colony, Tantalus V has really poor security. Van Gelder manages to escape with ease, by hiding in a box - the only box - beamed up to the Enterprise.
The Corbomite Maneuver - S1-E11
Other mistake: When Kirk is in sickbay with McCoy he uses sickbay's desktop monitor to contact Spock on the bridge, then when Kirk is in his quarters he uses his own desktop monitor to contact Spock again. Both have the same paint dings and smudges around the console screens, revealing the same one was used during filming. (00:04:30 - 00:06:30)
The Corbomite Maneuver - S1-E11
Continuity mistake: After they've blasted the cube, Kirk and Spock are discussing repairs on the bridge. Spock's earpiece is in place one minute, but vanishes the next when the camera angle changes. (00:13:30)
The Corbomite Maneuver - S1-E11
Visible crew/equipment: After the Fesarius destroys the recorder marker, when Balok declares that Enterprise must also be destroyed, in the next wide shot there's a tape mark on the floor near Bailey, and another one in front of Kirk's feet. (00:21:50)
The Corbomite Maneuver - S1-E11
Continuity mistake: Spock states that the Fesarius (Balok's ship) "must be a mile in diameter". Yet any one of the small spheres that make up the Fesarius dwarfs the Enterprise. If the Enterprise is about 300m long, the Fesarius would have to be around 6 km in diameter - considerably larger than a mile.
The Corbomite Maneuver - S1-E11
Continuity mistake: As Balok's 10 minutes are counting down, McCoy comes onto the bridge wearing a standard blue velour uniform. A few minutes later, when Spock says he can bring up Balok's image, McCoy is suddenly wearing his medical tunic with the different fabric and collar. Then when Bailey starts losing it, McCoy is back in the regular velour costume again.
The Corbomite Maneuver - S1-E11
Continuity mistake: As McCoy first appears from the elevator behind Kirk, then follows him in, his hairline changes from a widow's peak to straight across.
The Corbomite Maneuver - S1-E11
Continuity mistake: When the cube first meets the Enterprise, it's spinning clockwise on its axis. Later, it's spinning counter-clockwise.
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.