Character mistake: When Spock beams down after calculating there was just enough energy left for him to do so, Sulu and Kirk watch him materialize, and Sulu says, "Someone beaming down from the bridge." From the bridge? Shouldn't he have said "from the ship"? No one ever beamed down directly from the bridge, but even if they could, how would Sulu know that's where they'd come from? (00:28:00)
Character mistake: Kirk informs Spock via communicator the Horta is ten feet from him, and Spock insists Kirk kill it. First, both know they cannot kill it with their phasers, and second, Spock's demand for Kirk to kill the Horta runs counter to the Vulcan philosophy of respect for all life. Spock would never want to harm, let alone kill, another life form.
Character mistake: When Sulu is explaining the mathematics of Gary Mitchell's powers getting stronger, he says they are increasing "Geometrically", and then uses the popular example of doubling a penny every day for a month. What Sulu is describing would actually be exponentially. (00:24:14)
Character mistake: When Spock is finally able to determine that Edith must die to restore time, Kirk asks Spock how it will happen, and Spock tells him he can only estimate general events, not specifics. However, after Spock's previous attempt, he tells Kirk he'd seen her obituary. The obituary would necessarily include how and when she died.
Character mistake: When Kor describes the effects of the mind-sifter to Kirk (who is still in disguise as an Organian), he says the machine would leave him "more vegetable than human." There is no reason a Klingon would reference humans while speaking to a person he believed to be Organian.
Character mistake: In the scene when the shuttle is finally repaired and lifting off the planet, Spock asks "Fuel status?" and Scotty incorrectly answers "15 pounds PSI, sir. Approximately enough for one orbit." The correct answer would be simply "15 PSI" without the redundancy, as the P in PSI already signifies pounds. As an engineer Scotty would know this. (00:42:00)
Character mistake: Spock tells Kirk that without being able to test their vaccine with the ship's computers, "it could be a beaker of death." The vessel that the vaccine is contained in is a flask, not a beaker. The ship's science officer should know the difference. (00:37:25)