Is There in Truth No Beauty? - S3-E5
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Even if he did, humans can still go mad if wearing one and that's why Spock operates the transporter. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.
Rendezvous with Yesterday - S1-E1
Character mistake: When Tony talks to captain of Titanic he states he was born in 1938, but in episode 4 "The Day the Sky Fell In" it is December 1941, he should be 3 years old, but as a young child he is about 12 years old.