Bond arrives in La Paz and very soon afterwards is involved in fight scenes, car chases etc. In reality, as the altitude in La Paz is so great (12,000 ft), it is a struggle just to carry a suitcase out of the airport (even higher, at 13,500 ft), never mind to do all the things that Bond does. It takes several days to acclimatise before any strenuous activity is possible.
At the end of Casino Royale, Bond was wearing a light-blue shirt and a three-piece suit when he shot Mr. White in the leg. At the beginning of Quantum of Solace, his shirt is white and his waistcoat has disappeared, and in fact both he and White seems to be in slightly different suits. If the events of Quantum of Solace occur an hour after those of Casino Royale, it would mean that both had an inexplicable clothing change just before the car chase. (In fact, this continuity error betrays the fact that the films' costumiers are different: Brioni provided suits before, but for this film it is Tom Ford.)
When Bond is talking to Mathis at his villa, Mathis' watch moves between scenes. At the start of the sequence, his watch is facing away from the bottom of his wrist. By the end of the scene, it has moved to a more conventional position.
When Bond arrives back in Siena and M shows him the file containing the pictures of Vesper's boyfriend the file is open in order for Bond to steal a photograph and is then untouched until the end of the scene, when he pushes the file back across the table and it is already closed.
Heavy turbulence occur during the flight to Bregenz. In the single shot of Dominic Greene the water in the vase of white tulips is quite shaky. In the wide shot one can see that the water in the vase is not moving at all, before cutting back to the single shot with the moving water.
When Camille is talking with Greene while looking at the dead geologist, she wraps her arms around Greene's neck. Seen from behind, her arms are hanging at her sides, then back around his neck in the closeup.
In the opening chase, Bond's car and the villain's cars are black. In the quarry, right after the police's car is falling and crosses the track, we briefly see two white cars (or so covered in dust that they are uniformly white). A second after, they are black again (or with much less dust).
In the opening car chase sequence, the Carabinieri give chase in a Landrover 110, however, after it crashes into the building and falls of the cliff you can quite clearly see it has been swapped for a Defender 90.
When Felix Leiter is leaving the bar through the alleyway after meeting with Bond he is seen gesturing as if he's throwing something against the alley wall and breaking glass is heard, but the object (probably his beer bottle) has been removed.
After Bond shoots the first motorcycle police officer, he wrestles briefly with the second and throws him to the ground. There is a quick shot where he is shooting the second police officer, showing his motorcycle on the ground. The next shot shows the upright motorcycle falling due to the officer falling into it. It was edited so that you hardly see the motorcycle fall, but the way the officer falls, there's no way the motorcycle could have done what it did already on the ground.
It is clear that the outset of Bolivia was filmed in several different countries. Whereas in La Paz people speak with what appears to be a Mexican accent, people in the desert such as the desert hotel waitress and the plane owner speak with a Chilean accent. The Bolivian scenes were filmed in the Atacama Desert in Chile, Colon in Panama and the Baja California Peninsula in Mexico.
James Bond gets told on the phone that Greene's biz-jet is on a private charter to Bregenz, Austria, where he seems to land. However, Bregenz itself has no airport. Only St. Gallen-Altenrhein and Friedrichshafen are nearby (10 and 30 km, respectively). London Farnborough Airport was used for this scene.
James Bond is clearly not in Port-Au-Prince. The harbor is all wrong, there are no luxury hotels or home on the bay and the general conditions of the building are much better than in Haiti's capital. This was actually Colon, Panama, on the Atlantic coast.
During the opening car chase it is clear that this is filmed on the northern part of Lake Garda through the mountain passes and tunnels. After the quarry scene Bond is now suddenly driving into Sienna which happens to be around 150 miles from Verona, Lake Garda's main City location.
Bolivia is a landlocked country in South America without any access to the sea. In spite of that Dominic Greene, Camille and others talk in Bolivia's capital La Paz about the country like it has access to the sea, without ever mentioning that the sovereign countries Chile and Peru are between Bolivia and the sea.
During the aircraft "chase" scene the left engine of the C47/DC3 is shot out. Bond then sets both engines to full power and pulls back on the yoke to initiate a climb. Being that the left engine is the "Critical Engine" for the C47/DC3 Bond's course of action would not have resulted in a climb, but instead resulted in the C47/DC3 performing a roll to the left onto it's back and most likely crashing.
Critical Engine Definition per the FAA: The engine whose failure would most adversely affect the airplane's performance or handling qualities is the critical engine. On any airplane with both engines rotating clockwise the left engine is critical. Aerodynamically, this is due to asymmetrical thrust or "P" factor. The center of thrust is offset to the right side of the propeller hub, creating a greater distance (ARM) from the longitudinal axis to the thrust line of the right engine. This longer arm in turn creates a greater yawing moment, and with rudder deflection fixed at its maximum value, airplane speed must be higher to overcome the effects of "P" factor. Failure of the left engine, therefore, creates the more "critical" situation (need for higher Vmc)
Bond arrives at the opera and steals a 'goody bag' which contains the Quantum earpiece, he tips the bag into a sink in the men's restroom to examine the contents. The sink appears to have a tap (faucet) which operates automatically when hands are placed beneath yet the water does not flow when Bond is searching through the stuff in the sink.
While in Austria you see two license plates with 4 letters before the state emblem. In reality, all Austrian license plates have a maximum of 2 Letters before the state emblem. The emblem shown was the correct one for the state that Bregenz is in, but that letter combination wouldn't be possible.
M tells Bond that a DNA analysis has been done on a lock of hair. Strictly speaking, locks of hair are cut off and do not contain the hair follicle which is preferable for DNA analysis. However, it is possible to get DNA from hair samples that lack roots, although it takes more time and effort. It is likely that the case was given priority to facilitate the effort.
Bond uses a Walther PPK pistol throughout the movie. If Quantum of Solace is set immediately after Casino Royale, Bond would still be using a Beretta, as Q Branch is yet to have assigned Bond the Walther PPK. However, there is no reason an MI6 armorer could not have switched Bond from his previous Walther P99 to the PPK before he shoots Mr White at the end of Casino Royale. We do not see his sidearm, only the H&K UMP he uses to kneecap White (which he still has in Quantum). He could have had said PPK on him the whole time.
It would be highly unlikely that in any country René Mathis could be charged, tried, acquitted and compensated in the time between the two films. However, there is no indication of how much time has elapsed between Bond getting Vesper's text message with Mr. White's phone number and Bond finally tracking him down. Also, Mathis was taken into MI6 custody for interrogation, not for trial.
Though the film is seen to follow on almost immediately after the events of Casino Royale and yet things such as M's office at MI6 HQ are completely different, it is clear than the final scenes of the previous film take place some considerable time after Bond finds Vesper's clue to lead him to Mr White. M has come to Italy and the sting has been set up, with a new Aston Martin and the safehouse in Siena, and White's extreme rendition arranged.
Bond gets Elvis' mobile number and the company name Greene Planet by tricking him with a Universal Exports business card - the company's logo comes up on Bond's phone when Elvis tries the number. When Bond namechecks Dominic Green with MI6 not long after, however, having got this name from Camille, the name 'Greene Planet' seems to mean nothing to him, and he does not confirm Greene's identity until he receives an image.
We learn from M that Dominic Greene was found dead in the desert with two bullets in his skull and motor oil in his stomach. Presumably Greene drank the oil, and was then shot. Why would Green's murderers kill him if he had just did that to himself by ingesting oil? If Quantum was trying to make a lesson out of Greene, why not let him writhe in agony rather than deliver a coup de grace?
Bond and Mathis are seen flying on Virgin Atlantic between Europe (presumably Italy) and Bolivia. However, the airline has never offered flights to South America, and all of its flights originating in Europe depart from the UK. (Virgin Atlantic is, however, a longtime sponsor of Bond films, and its founder, Sir Richard Branson, has had a cameo appearance in at least one film.)
The goof items below may give away important plot points.
Bond has a bloodstain on his shirt when Mathis is dying in his arms. After Mathis dies and Bond disposes of the body, his shirt has no bloodstain on the lower front. Immediately after, when Bond is renting the plane, the bloodstain is back.
When Mathis is dying in Bond's arms, the amount of blood on Bond's right hand changes between the three different shots, from very little in the long shot, to none on his thumb, to blood covering his whole hand in the close-up.
After Bond puts Mathis' body in the dumpster, Mathis' left hand is hanging over the edge of the dumpster. In the next shot when Bond is going through his wallet, his hand is no longer hanging over the edge. In the following overhead shot of Mathis, his hand is once again hanging over the edge of the dumpster.
During the burning compound scene, Mr. Green's sidekick Elvis is standing with gun drawn poised to shoot Bond. When a fuel cell explodes we see him standing as the semi transparent flame hits him. For a few frames it appears his clothes have blown off but in the reflection on the wall near him, the pants are still on.
At the end, when Bond and Camille are trapped in the burning hotel, he shoots a fuel tank; in the movie, the explosion blows out a wall and leaves them unscathed. In reality, explosions follow the path of least resistance and would have affected them before having any impact on the wall. In a best-case scenario, it would have left them alive but severely burned, critically injured, and deaf; more likely, it would have killed them.