Revealing mistake: Just before Bond and Natalya are ejected from the Tiger, a mid-range shot reveals a pair of dummies in their place. (01:12:15)
Factual error: The missiles launched by the helicopter when Bond and Natalya are tied up in the cockpit are Mistrals, as revealed by the cockpit's screens - short range infra-red homing missiles. They home in on heat, they can't be programmed to turn 180 degrees in mid-air and target a specific location as they do here.
Other mistake: It's quite convenient for the plot that Natalya knew about both Goldeneye satellites but never seemed to know where Mischa was operated from otherwise she and Bond could have gone to Cuba without having to spike Boris' location.
Plot hole: When Alec reveals himself to be Janus and that he was alive all that time Alec scolds Bond for re-setting the timer at three minutes instead of six. The problem is if Alec had realised Bond reset the timer on the explosive then he should have had plenty of time to leave the building after Bond escaped, so he shouldn't have been injured.
Suggested correction: At the time he DIDN'T know Bond had reset the timer...maybe Bond had agreed to do this off-screen at some point but it's never alluded to.
Visible crew/equipment: At Q's lab, when Bond opens the car door a huge round stage light is reflected, which wans't there before in the previous opposite angle.
Continuity mistake: The first time we see St. Petersburg the amount and type of people behind the soldiers parading changes between shots.
Continuity mistake: Xenia's hair is tied up in a pony tail, but when Bond fires at the helicopter it's loose. Then it goes back to a pony tail. (01:43:00 - 01:44:01)
Revealing mistake: When Xenia dies and the helicopter is about to explode, the pilot is replaced with a very obvious stiff dummy which falls forward without control.
Revealing mistake: Bond's stunt double is incredibly visible running up the stairs at the antenna, while being shot at by 006. The double is visible again when Bond jumps down the stairs and rolls. (01:58:10)
Character mistake: At MI6, when M is showing Bond a profile of Ourumov on a screen, Arkhangelsk is misspelt as "Arkhangeesk." (00:46:36)
Revealing mistake: When Bond kicks Trevelyan off the antenna ledge, the side-on shot reveals both stuntmen instead of Pierce Brosnan and Sean Bean. (02:02:41)
Revealing mistake: In a close-up of Xenia rapelling down from the helicopter, you can see that Xenia is not Famke Janssen but a stunt double.
Factual error: When Bond and Natalya slide down the dish after being shot at, Bond stops them by putting his feet against a bar or something at the edge of the hole. However, they are traveling so fast that this should have broken (or at least somehow injured) his legs. Plus, that little thing called inertia should have sent them both over the bar, down the hole, and to their deaths. (01:48:15)
Audio problem: Near the beginning, Bond is racing another woman on the mountain. The woman in the car with Bond says she enjoys a spirited ride but her mouth doesn't match what she is saying.
Suggested correction: Checked, and the audio matches her lip movements just fine.
Other mistake: When Xenia's helicopter crashes the pilot vanishes.
Character mistake: When Boris teases Natalya about not telling her the password to his spike program, she glances down at the keyboard as he types the mystery password ("chair") and should've been able to make it out, given her level of programming expertise. (00:30:26)
