Deliberate mistake: When Eddie and "Jennifer" are walking around the house alone, she is trying to tell Eddie that she's really Sara. But by the time she is explaining that, Eddie has disappeared. She doesn't find him until later. But from the moment she finds him (and he is talking to her) he calls her Sara, despite never actually hearing her say that. Also, after they all get back together, a few of the other people start calling her Sara as well, but no one actually heard that either.
Deliberate mistake: While Hector is in the water with the crocodile, he yells to the female officer to start the helicopter by turning the key. It takes a lot more than just the key to start a Bell Jet Ranger helicopter. Fuel shutoff valve, throttle, key, and the main rotor doesn't start rotating until the throttle on the collective stick is turned to a specific position.
Deliberate mistake: The shot of Ardeth and his companions watching Rick leave Hamunuptra at the start of the film is reused again when Rick, Evie, Jonathan and the Warden re-enter Hamunuptra, although the reused shot has a blue filter over it. Ardeth even says exactly the same line he says in the original shot, but it is subtitled differently.
Deliberate mistake: When Jojima's car is blown into the air, one of Ryuuichi's thugs throws a knife at it. When Jojima gets out after the car crashes to the ground, the knife is stuck in his stomach. This is physically impossible, as the knife would have to go through a large portion of metal to hit him there.
Deliberate mistake: The currency conversion program Bond uses to convert GBP into USD only has fields for a 7 figure amount. That's rather small for an organisation like MI6.