Deliberate mistake: Blofeld doesn't recognize James Bond in this film, even though they met face-to-face in the previous movie, "You Only Live Twice." There is a production-related reason for this. Ian Fleming wrote "On Her Majesty's Secret Service" in 1963 (in which Bond and Blofeld met for the first time), and he wrote "You Only Live Twice" in 1964. However, "You Only Live Twice" was adapted for film first (in 1967), and "On Her Majesty's Secret Service" was adapted afterward (in 1969). Because the 1969 film was so faithful to its source material, Blofeld and Bond are basically meeting for the first time...again. The producers were aware of this continuity problem and intended to have James Bond undergo plastic surgery for "On Her Majesty's Secret Service" (which would conveniently explain Blofeld not recognizing him, as well as the fact that Sean Connery had been replaced by George Lazenby in the lead role). But the plastic surgery idea was discarded in faithfulness to the novel, resulting in a glaring continuity problem between the 1967 and 1969 films.
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. (02:01:25 - 02:02:40)
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.
Continuity mistake: When Tracy is battling that balding guard during the climax, she finishes him off by pulling him into a metal patterned wall in a way that ends up with one of the circular sections on his head like a crown. When he falls down the stairs this circular section disappears. (02:02:00)
Other mistake: In the scene when the helicopters arrive over the research center, look at the heli in the bottom right - it just fades out, while one on the left materialises, and the one in the middle changes position. Notice that after all the helicopters have changed place, you can see the helicopter that faded out appear on the helipad (02:25:25)
Continuity mistake: In the stock car race scene, when Bond and Tracy first drive onto the track the number 11 red car spins out to avoid a collision and ends up facing against the wrong way. Two seconds later we see the number 11 car backing up so as to face the correct direction. Two seconds after that, in a long shot, we see the number 11 car going forward again. A few seconds later we see the number 11 car facing the wrong direction as it begins backing up again. Several seconds latter, the Mercedes containing Blofeld's henchmen enters the track and once again the number 11 car is stopped facing the wrong direction. Later in the scene the Mercedes collides with the number 11 car causing it to spin out and end up facing the wrong direction, yet a few seconds later we see the number 11 car racing along in the correct direction as Bond and Tracy pass it.
Factual error: Of all the characters, major and minor, who ever speak German in Switzerland, not one speaks Swiss German. The accents are very different (Swiss German can be almost incomprehensible to speakers of High German), and there's also different vocabulary. While some of the characters might just be non-Swiss speakers of German, at least the underlings at Piz Gloria should be speaking Schwyzerdeutsch.
Continuity mistake: In the car chase in Switzerland Bond and Tracy are being chased by Blofelds men, and at one point the Mercedes with the bad guys swerves off the road. But in the inside shots of Bond and Tracy's Cougar you still see the Mercedes' headlights, still on the road, in the rear window.