Character mistake: In the ice-hockey scene Bibi says: "They say that he is a defector from East Germany." That fits very poorly with the statement during the biathlon scene, where he is referred to as "The East German Champion".
Revealing mistake: Bond kills Locque by pushing him over a cliff in the Mercedes. When the wrecked car finally comes to a halt upside down at the bottom of the cliff you can see that the running gear and engine of the Mercedes have been removed and the wheels are simply held on with 2 straight beams (it is a different car altogether, just a shell with wheels).
Revealing mistake: When Locque kills Countess Lisl by running her over in his dune buggy, look in the cockpit and you can see that the driver is not Michael Gothard but his stunt double.
Continuity mistake: In the scene when Bond shoots Locque's car window and the car is hanging over the edge of the cliff, Locque puts his hand on the door when the car starts to fall, but in the next shot when Bond kicks the car over the cliff, his hand is no longer on the door.
Continuity mistake: When Bond is fighting the diver over the ATAC, Bond is holding onto it with his left hand. The diver then sends Bond crashing through a glass window, and when it cuts to the next shot, Bond is now holding onto the ATAC with his right hand.
Revealing mistake: Kristatos wants to run over Bond and Melina by boat, after he took the Atac. Bond and the girl swim to the sea floor, waiting till Kristatos is gone. You can see the (shadow of) the anchor hanging on the left side of the ship. But in the previous scene, you will notice the anchor is in the middle, on deck! And second: these kind of (speed)boats do not have an anchor attached below the water line, because it will reduce the speed. Conclusion: the underwater scene is fake.
Continuity mistake: When Bond is climbing up to the mountaintop fortress, he hammers in pitons as he goes up. But when he falls down, the three pitons are all level.