Fast & Furious takes place 5 years after The Fast and the Furious. Assuming the first took place in the same time period it was made (2001), then this installment would take place circa 2006. Brian driving a 2009 Subaru STI is therefore impossible.
When Brian is choosing cars out of the impound lot, he is seen choosing several Nissan GT-R chassis for rebuilding into a car to be used in the street race to work with Baraga. One of the cars chosen is an R35 GTR. As the movie takes place 5 years after The Fast and the Furious, this would be impossible, as the first GTR was not seen on US soil until 2008, two full years after the movie takes place. This continues into Fast Five - there is a GTR featured in that movie as well, however, The Fast and the Furious: Tokyo Drift takes place in approximately 2006, which is also after the events in both Fast & Furious and Fast Five.
When Brian is under a bridge, talking on the phone with agent Penning after the hijacking of the Hummer with 60 million dollars-worth of heroin, it is dark. When he heads to the back of the Hummer to see where has Dom gone, it is sunny.
Torreto's Buick Grand National has the unmistakable sound of a V8 when in fact all GNs had a turbo V6. Obviously a V8 could have been transplanted but then why go to all the trouble to make this car cosmetically correct as a Grand National (right down to the Grand National logo on the front license plate)?
In the beginning scene after Dom asks Letty to jump, you can see that the truck driver jumps from the driver side to the side of the road. In the very next shot, the truck is on its correct lane, so it wouldn't be possible for the driver to jump that far.
During the scene when Dom slams his Buick into the tanker hitch, the quarter-panel is shown at first to be damaged and then a shot later, not damaged then back to damaged again which could also be related to another goof reference to the GN's grills being 2 different colors in different shots; they used 2 different cars.
When O'Conner is racing with Dom to see who gets the spot for Braga, in the final part of the race a car falls of a bridge and lands in front of Brian's car with the font part of the car into the ground. With that speed the car would have fallen a bit further. and between two narrow shots, the car while falling changes it's position relative to the bridge.
When Letty is speaking with Dom on the beach, her loose shirt appears to be in a variety of positions on her shoulders in between shots. Sometimes it's on properly, other times it's falling off the left or right shoulder.
When Dom is at Brian's side in the desert near the end of the movie, they are in a vast, flat desert. When the camera pans out to show the police cars driving towards the two, the scenery dramatically changes, and resembles a Coastal Plain with grass on the ground.
David Park's car is described as a '98 Nissan 240 with illegal mods', which would make it an "s14". Though it is actually a Nissan Silvia S15. S15's were never produced for North America, while S14s were, which would mean it was imported from Japan. The body lines on the rear quarter panels, front end (headlights, fenders, etc.), and taillights prove it's an S15. It's also right-hand drive.
When O'Connor is going after the criminal, the criminal is shooting with a six chamber revolver. However, when he takes the firsts shots at Brian he fires 3 bullets. When Brian pushes him off of the building he fires another 4 bullets. That's 7 bullets in a six chamber gun.
When the young boy in Panama tells Torreto he has a phone call, Torreto rolls out from under his car holding a water pump. The water pump in that particular car is not accessible from underneath the vehicle.
In the beginning they unhook the gas trailers and reconnect them to a tow truck. Simply unhooking a tractor trailer and dragging it away would not work as the air line needs to be connected for the brakes to release.
Agent Penning says with regard to Arturo Braga that "he's moved more heroin across the border than Escobar did in 10 years." This references the notorious Colombian drug cartel leader Pablo Escobar, who rose to power in the 1980s. However, Escobar and his Medellin cartel primarily trafficked in cocaine, not heroin as Penning's quote would imply.
Dom blows up four cars by pushing the cigarette lighter in and releasing nitrous oxide into the car. Nitrous oxide is not flammable or explosive at room temperature, however, just because it's a NOS tank doesn't mean it actually contains nitrous oxide. It's likely that Dom had planned beforehand and filled the tank with a flammable gas.
Dominic's Chevelle has narrow wheels while in the race to make Braga's team. It may be more difficult for him take the corners with drag front wheels but not impossible even though the car was set up for a straight line, and not cornering.
During the end credits a car's engine is shown firing in cutaway. The graphic indicates a combustion event on every stroke of the piston. While not common some cars with custom engine management systems run 'wasted spark' firing. In this system there is a spark every compression of the piston.
Brian's (Paul Walker) reason for not turning in the money to the FBI after the drug run is that Braga will just continue to run drugs without the loss affecting him. In the same conversation, when told by his boss that Braga will not show up for the exchange, Brian tells him that he can't afford not to.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
In the final scene where the crew is going to get Dom out of the prison bus, there is a Honda NSX that is left hand drive. This means that this car must be a Honda NSX From Europe where the Honda NSX was sold as a left hand drive, however in the end credits it is shown that the NSX has a V8 but this car was only produced with a V6.