Plot hole: Blofeld tells the bad guys that they had Bond and let him get away. The woman says, "But Bond is dead. It was in all the papers." As we saw in a paper near the start, there was a well-publicised photo of Bond, and when she met him he wasn't using any kind of disguise, so how did she not recognise him if she'd seen the papers?
Continuity mistake: There's a scene where Audrey Hepburn's husband Sam is leaving their apartment. Standing on the pavement outside he hails a cab, which pulls up through a (quite sizeable) puddle, splashing him with water. Opposite him on the other side of the road are parked two cars and the villains' Volkswagen van. There's a quick cut-away exchange of "pleasantries" between Sam and the cabby before it drives off. The next shot has the villain (Richard Crenna) emerging from the van, crossing the street, and making for the door to their apartment, only all of a sudden the puddle has disappeared, as have the two cars which were parked behind the Volkswagen van.
Plot hole: At the beginning of the movie, Reese (and Walker) rob from a "drop" at Alcatraz. Reese uses that money to pay off "The Organization." At the end of the movie Brewster tells Walker the only place to get cash is at the drop. "The drop is the same but the location has changed." Brewster receives the money at the drop. This makes it clear the original drop was stealing from "The Organization." Why would Reese be able to pay off "The Organization" by stealing their own money from them?
Revealing mistake: Inspector Emil arrives at the morgue; pay attention to the body, just before he asks "You are...?" to Armstrong, the sheet covering the cadaver rises and fall as the departed takes a few deep breaths. (00:32:00)
Continuity mistake: Alain Delon tries on some jackets. He finds a briefcase and opens it. You can see the contents of the briefcase before it cuts to a closer view; in this closer view, a stack of letters moves, and an orange case vanishes entirely. (00:27:00)