Continuity mistake: When the car chase enters a car park there is a blue car in the bottom left corner (from the overhead shot), next to the trailer which Bond makes his escape over. After Bond escapes the car is white (I think) and may also be pointing the opposite direction.
Visible crew/equipment: When Bond and Tiffany are talking while looking at the fish, he suddenly rolls her on her back, saying "I'm on top of the situation". As she turns you can clearly see a plaster (band-aid) carefully placed to cover any...er...accidental revealings.
Plot hole: When Bond first sees that Willard White is actually Blofeld in the hotel penthouse, Blofeld shows him how he has been impersonating Willard White's voice with a machine. But that only works through the phoneline - out loud it would be different.
Revealing mistake: During the car chase, Bond weaves through a car park, and a police car can't make the turns, sliding into a wall. The wall's covered in one specific point by what looks like a large pale wooden cover, which the police car just happens to slam into exactly.
Continuity mistake: In the battle scene on the oil rig, Bond takes takes control of the crane attached to Blofeld's bathosub. During this sequence the folding window at the front of the crane cab switches between the raised and lowered position several times. (01:47:53)
Factual error: When Bond is inside the oil rig, the shots switch from the satellite control room to the satellite itself orbiting. When the satellite fires at the underground missile, it's orbit direction reverses. Then, when it fires at the Russian sub and the Chinese military base, it stops before it fires. This is impossible, as Newtons 3rd law states that for every action there is an equal and opposite reaction. For the satellite to reverse or stop, boosters would be needed at the front of the satellite (of which there aren't any) to counter it's speed. And even if they were there, the direction change and stop wouldn't be as sharp as the movie shows.
Factual error: When Mr. Wint and Mr. Kid assassinate the doctor in the South African desert, they use a scorpion. Scorpions are poisonous, but the sting is more like a bee's, and woudn't kill a healthy adult.
Revealing mistake: Blofeld is played by an actor that is not bald (Blofeld was bald in 2 previous episodes). He is also played by Charles Gray, who played the part of one of the good guys in You Only Live Twice. [Not really a mistake, given that Blofeld doesn't have to conform to certain physical characteristics. However, an actor playing a good guy then a bad guy over the space of two films rankles a bit...]
Revealing mistake: When Saxby is shot, he has an instant, pre-dried blood splatter on his shirt.
Factual error: The satellite uses a laser as its weapon, which can be only as powerful as its root energy source. It obviously doesn't have a nuclear reactor large enough to yield that much power, and it can't be solar power, since the collecting area of the dish is orders of magnitude too small to provide power at a level sufficient to lay waste to whole submarines and missile installations.
Audio problem: In the final scene, Mr Wint and Mr Kid bring in a dinner trolley. When Mr Kid places the desserts top over the bomb, we can hear the timer ticking. Yet, when the dessert is thrown and the bomb is revealed, the ticking cannot be heard.
Continuity mistake: At one part Bond is talking on the phone. He is wearing a pink tie that goes to the second button on his shirt. Then in the next scene before he meets up with Thumper, his tie is longer.