Revealing mistake: When James Bond and Felix go out in the helicopter for the second time to search for the mising plane, at the moment when Bond is looking through the pair of binoculars, and when Felix says "you won't find it down there, that's the Golden grotto" just look how slowly the rotor blades are turning, at this speed there is no way that the helicopter could possibly fly and besides, there is hardly any noise from the engine.
Continuity mistake: One of the gadgets which Q gives to Bond is a mini air tank, "good for about four minutes of air." He's very specific about this number. In the huge climactic underwater battle scene, Bond Against The Hordes, 007 puts on the mini tank and proceeds to fight, vigorously, for a solid ten minutes without borrowing anyone else's tank. He either had gills implanted or can fight while holding his breath for enormously long periods of time.
Plot hole: It would not have been possible for Bond to place his jet-pack on the roof of Jaques Bovier's mansion. If he flew up there, everyone would have heard him - it's very noisy. And since it's also quite heavy, it is hard to imagine that he would have time to climb the wall, place it there, and get into the mansion, during the same time it took Jaques Bovier to get back from the church - which he left before Bond did.
Factual error: When the operation to plant the bombs is foiled, the Disco Volante casts off, pursued by two speed-boats and a destroyer. But how could the latter, as a large (and slow) battleship, approach the yacht without arousing suspicion?
Continuity mistake: When Bond and Felix go out for the first time to search for the missing plane, Bond's hair has a parting to the left and is wearing watch on his left arm, but at the moment when they are flying over Largo's house, Bond is looking through a pair of binoculars. As he says to Felix "separate pools for sharks, no less" his hair is now parted on the right and his watch is now on his right arm, also watch out for the mouthpiece on Felix's headphones - it changes from left to right.
Revealing mistake: When Bond is wounded and on the run from Fiona and her assassins, he tries to lose them in a parade. At one point during the chase, for several seconds, there is a dog in the frame. Even without slowing down the film one can see that the dog is stuffed.
Revealing mistake: When the water jets are turned on in the DB5 you can see the fire hoses attached to the car providing the water.