Continuity mistake: When Tracy and Bond are talking in the casino, and Bond tells her to stay on 5, she has her left hand resting against her chin. As she says "people who want to stay alive play it safe" she moves her hand down to the table, but then when the camera cuts back to Bond the reverse shot of her shows her hand still by her chin, and then she moves it down again.
Continuity mistake: At the start of the scene where Bond is helping with the assault on Piz Gloria, look at Tracy's hand just after the bald guard presses it down on the object to disarm her. You can see blood, but a couple of minutes later when she kills him there is no blood to be seen.
Plot hole: Blofeld is played by Telly Savalas. In a previous film, Blofeld had met James Bond and had ordered him killed, yet in OHMSS, when Blofeld meets James Bond, he doesn't recognise him. It's well established over the course of the Bond movies until the reboot with Casino Royale that they all take place in one timeframe, so they should have recognised each other.
Other mistake: In the bobsled chase, Blofeld arms a grenade, which drops to the floor of his sled, where it rolls around with every turn. He arms and throws a second grenade onto the track, where it is detonated by Bond's sled. The original armed grenade loose in Blofeld's sled never detonates, even when that sled crashes.
Plot hole: Blofeld tells Bond that he realised he was not the real Sir Hilary Bray when Bond told him that the Bleuchamp tombs are in the Augsburg Cathedral. Blofeld informs Bond that they are in-fact in the Saint Anna Kirche, and the real Sir Hilary Bray would've known this. However Bond had gotten his information about the Bleuchamp tombs in Augsburg from the real Sir Hilary Bray earlier in the film.
Continuity mistake: When Bond is fighting the villain on the beach in the pre-credits sequence, the two fight in the sea and Bond's hair gets wet in the process. But when Bond says "This never happened to the other fellow" his hair now appears to be completely dry.
Continuity mistake: When Bond and Che are fighting, Che knocks out one of the white wooden bars behind Bond. However, in the next shot, two bars are missing on the upper half of the wooden structure, one on the bottom half. Seconds later, just before Bond throws Che into it, the bottom half of the structure is complete again, and only one bar is missing on the upper half.
Visible crew/equipment: When Bond is being chased on skis, he flies over a ridge and through some trees, landing very close to the edge of a cliff. A cable is clearly visible tied to his right ankle to keep him from going over the edge.