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.
Continuity mistake: Phelps takes the tape from the recorder's right-hand spindle. When he opens the acid vat, we can see the tape he just removed back on the recorder - but on the left spindle. The tape he then drops into the acid doesn't remotely resemble the one he pulled, and when he drops it, the recorder's left spindle, where the full tape was in the last shot and where the empty reel should be, now has nothing on it at all. (00:02:50)
Visible crew/equipment: Oliver (a dog) jumps up on Uncle Bill's bed and starts barking at him. The real reason the dog is barking: in the shots of Oliver, the shadow of a towel or some sort of toy can be seen being waved around in order to agitate the animal. It even briefly pops into the shot a couple of times. (00:10:45)
Continuity mistake: Part 3: As you see the third "Sleeper" (a giant bomb shaped like the Red Skull's head) note the position of the monocle. First it's on its right side, then its left side, and back again to its right side. (00:14:20 - 00:15:05)