Continuity mistake: When the bathroom explodes, from one angle we see the toilet come sailing out over the roof of a police truck parked at the front of the house. While the angles make it a bit tricky to judge, there's a definite leftward trajectory. We then cut to a closeup of Murtaugh's car, and the toilet lands on it as if it was flying to the right. Even if the angles are misleading, in the first shot the police truck and other cars are parked to the right of the exploding window. In the shot of Roger's car he's also parked to the right, but without all the cars visible around it which there should be. (00:57:15)
Suggested correction: The toilet is flying straight out and straight towards the car - you can see this from the perspective of the interior of the car prior to the toilet becoming visible. Next the toilet is flying towards the area in front of the police van, not over it. And the car is parked in front of the van. When we see the car again, the van and other police cars are out of frame, so they can't be seen.
The mistake is valid. When we see the toilet flying, it's moving to the viewer's left, but when it hits the car, it's moving to the viewer's right now. Even if in the first shot, with the angle of the camera, the toilet is flying straight, it doesn't hit the car straight on. And he's not saying the toilet sails over the roof, but that you can see the toilet in the background is above the roof of the van in the foreground and that's moving over it (from right to left).
Continuity mistake: When Rika comes to Riggs' trailer he offers her a chair, and she sits down twice in different camera angles. (01:07:50)
Suggested correction: This is too subtle to be called a mistake - she's in the process of sitting when she's facing the camera, the shot changes to behind her and she's seen finishing her settling into the chair - this could just as well be her making a natural adjustment to her sitting position. It's not an obvious mistake by any means.
Other mistake: When Riggs kills the guy who murdered his wife by dropping that cargo crate on him it's obvious the crate has no bottom and it falls around him, not least because the body's height doesn't change as the crate falls, there's no sign of impact or crushing, he just stays perfectly upright as it hits the ground. (01:39:25)
Suggested correction: This is simply not a mistake - we obviously know that the filmmakers didn't really crush an actor, but there's no visible clue of the SFX "deception."
Continuity mistake: When Rika sits down outside of Riggs' place, the front of the trailer is half shaded by the sunroof. When the camera angle changes the shade is gone, and it's much brighter. (01:07:50)
Suggested correction: This entry is only half right - the "sunroof" or awning is visible from both camera angles, though it's true that the sun is lower and is lighting more under the awning when the camera moves inside the trailer.
Factual error: Diplomatic immunity does not give a foreign diplomat carte blanche to openly and brazenly commit extreme crimes in their host country. Arjen Rudd and his men are known money launderers. Once Riggs begins harassing Rudd at the consulate, Rudd launches a violent campaign against the Los Angeles Police Department, assassinating half a dozen police officers. At the very least, the United States would be well within their rights to expel Rudd from the country and bar his re-entry. There is no way the South African government would oppose prosecution of Rudd given these circumstances, doing so would fracture all diplomatic relations with the United States. It is absurd for Rudd to shoot a cop and smugly proclaim "Diplomatic immunity!" after he has done so because it is not a "get out of jail free" card. If Rudd were expelled, he could face prosecution in his home country.
Suggested correction: Before the federal government would declare a diplomat persona non grata, which would be a serious degradation of the diplomatic relations between two countries), they would first have to do an investigation. The entire movie seems to take place in less than a week. There was probably not enough time, especially since Riggs and Murtaugh are playing things close to the vest and not letting people know what they are doing. As for the "Diplomatic immunity" line, he's just being insulting towards Murtaugh.
While it is true that it's difficult to actually expel a diplomat, the mistake is that Rudd acts as if it is impossible. The movie operates as if diplomatic immunity makes it impossible to arrest a diplomat. And Rudd is doing more than just taunting Roger, he believes right up until the end that he can't be held accountable for his actions, up to and including murdering a cop.