Continuity mistake: The aging Kirk's hairline has receded considerably. When he ages still more and goes completely gray, however, the receding process inexplicably reverses itself, and he suddenly has much more hair on his forehead than before. (00:34:35 - 00:43:55)
Factual error: As Kirk, McKoy and Scottie age during the episode, their hair goes from brown to fully grey. Mr. Spock shows this to a lesser extent, but by the end of the episode, he has full streaks of grey. This isn't how grey hair works. It starts from the root and works it way outward over time to the end of the strand. Unless their hair is growing quite rapidly, and then cut to keep it short, they shouldn't have full heads of grey hair.
Revealing mistake: While Lieutenant Galloway is coming into sickbay and dies there is a window in the hallway with window blinds, and through this window you can see some of the production staff in regular clothing talking.
Continuity mistake: Mr Checkov has a rather unfortunate hair styling during the beginning of his role in Season 2. In this episode his hair is normal until the end, where in between being given an order and carrying it out he somehow manages to go and have his hair professionally done and made all bouffant. (00:47:15)