Deliberate mistake: So that he can look in the right direction, toward Lincoln, a shot of Kirk is reversed before the first battle, putting his uniform insignia on the wrong side. (00:25:45)
Revealing mistake: When he kicks Colonel Green away during the first fight, Kirk splits his pants. Fortunately, they mend themselves a few shots later. (00:28:00)
Visible crew/equipment: When Kirk is agonizing over the sound of Surak's supposed cries for help, an equipment shadow, possibly from a boom mike, skates across the rocks behind his left shoulder. (00:40:00)
Continuity mistake: Lincoln asks Kirk if they measure time in minutes and Kirk replies "We can convert it." In every previous episode they've always used minutes - their ship chronometers are still set to minutes.
Revealing mistake: When Kirk & Spock beam down with Lincoln to the supposedly newly-created non-volcanic landscape, there are footprints all over the ground. They disappear a few shots later, though.
Continuity mistake: During the hand-to-hand fight scene, the stunt double playing Spock gets his hair mussed up, but immediately after the fight, Spock's hair is neatly combed and formed, as it was before the fight.
Revealing mistake: When Abraham Lincoln shoves the Klingon away from him during their fight, the Klingon collides with a heavy boulder which moves several inches at impact.
Revealing mistake: About 4/5 through the show, while the Vulcan is being killed, if you look at Lincoln's wrist, you can see his makeup line.
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.