Revealing mistake: During the gun battle after Renard escapes the bunker, someone gets shot, and you see the stunt bloke dive to the floor and scream even though the gun wasn't even pointed at him or any bullets fired. He's the one standing just to the right of the jeep as we look at it.
Suggested correction: He's jumping out of the way, so he doesn't get run over by the jeep.
Continuity mistake: Right at the beginning of the film when Bond is trying to escape from the banker's office, he goes to the window and tucks his gun into his waistband. He then runs to the body lying on the other side of the office, and tucks it into his waistband again.
Suggested correction: This is incorrect. I've just watched the scene. Bond has his gun in his hand when looking out of the window and seeing police cars arriving. He then tucks it into his waistband to try to open the window. After the unseen sniper shoots the guard and Bond dives for cover, you can see his right arm pulling the gun out of his waistband, as he tries to work out where the sniper is. Then once he runs over to the dead body he tucks his gun back inside again. Not a mistake.
Other mistake: When Bond is about to jump from the banker's window, he is shown tying a knot on the strap, but when he gets to the street, he only releases the strap and walks away.
Suggested correction: He may have tied a quick release not, which by sliding his hand back up would have released the cable.
Bond does not tie a quick release knot. He ties a traditional overhand knot (it's shown in quite some detail), which cannot be easily removed. Therefore the original entry still stands.
Continuity mistake: The gun that Elektra uses near the climax and which Bond takes from her is a Colt Mk IV Series 70. However, when he's on the submarine, it turns into a regular M1911A1. Compare the finish on the wood grips and the fact that the Series 70 has a stainless barrel, whereas the 1911 doesn't.
Suggested correction: There are a lot of bad guys on the submarine, it's entirely possible that Bond has killed one of them between shooting Electra and pulling the gun out again. chances are he may have even lost it when he dived to the sub.
I went back and checked this. The sequence of events after killing Elektra make a point of showing him tuck the gun in his trousers. He dives, swims to the submarine, and interrupts a sailor that is closing the hatch by kicking this man in the face and sneaking onboard. Literally the next shot after, a bald bad guy is walking down a corridor towards the camera and the gun comes into frame (held by Bond) and stops the bad guy in his tracks. Note the gun that is shown is drenched in water and it is the wrong gun from earlier at Elektra's. The hammer is not stainless steel and it is missing the gold insignia on the grip. You can even kind of make this detail out as the gun is shown still tucked in Bond's pants when he makes it down the ladder when entering the sub. It seems to already be the wrong gun. Besides the gun being covered in water, we just saw Bond kick a bad guy in the face and just a second after in the next shot he's holding up this second bad guy. To assume he needed to switch guns in between this very brief cut makes little sense when we just saw he had the gun tucked in his pants from Elektra's.
Revealing mistake: Near the end of the movie where Renard and Bond are going at it, Bond is choking Renard and vice versa. You can clearly see Bond's thumbs aren't even touching Renard's neck.
Continuity mistake: When Bond tells Renard that Elektra is dead, in the over-the-shoulder shot of Pierce Brosnan, you can see him making faces at Robert Carlyle (just before he starts screaming). However in the next shot of Bond's face, it's as if his face hasn't changed.
Suggested correction: The close-ups of Renard do not make it clear enough to see what Bond's expressions are.
You can see his face to the right of the screen mouthing something.
Revealing mistake: During the action sequence on the labyrinthine docks on the Caspian Sea, one of the choppers with the rotating blades saws Bond's BMW in half. A cutaway shot reveals Bond nearby, murmuring, "Q's not gonna like this." The next shot is in back of Bond, seeing the newly split BMW fall apart, but Pierce Brosnan has been replaced with a dummy. This is most obvious during the over-the shoulder shot of the BMW falling apart, where Bond's gun is now held in front of him. Also, when the car splits in two, the dummy shakes a bit.
Suggested correction: The over-shoulder shot is not clear enough to reveal a dummy; when the car splits in two, the whole boardwalk around it shakes, as would the real actor/double.