Continuity mistake: When Hunt reverses the truck into the street and drives off, dumping the bad guys onto the road, the doors are swinging about as they should be, but for the rest of the chase they are shut and latched so they can't come open again. This has to be done manually.
Continuity mistake: When Ethan is thrown from the helicopter, he lands with a straight black strap attached to his left wrist. When he comes to a few seconds later, the strap is now doubled up and he's gripping it, it's not attached to him. (02:03:30 - 02:04:30)
Continuity mistake: When Hunt and Walker extract Lane from an armoured convoy by alternate means, right after they push the truck containing Lane into the water, one of the police in the convoy tries to shoot Hunt and Walker, leaving 3 bullet holes in the front windshield of the truck. Afterwards, the bullet holes keep disappearing and reappearing when Hunt and Walker escape with the truck across Paris. The most obvious occurrence when the bullet holes disappear is when Hunt and Walker kick the front windshield of the truck to flee with a motorcycle after it gets stuck between buildings.
Factual error: After a helicopter warning signal says pull up, Ethan pulls back on the cyclic control. All that would do is slow the helicopter by changing the blade angles. He should have pulled up on the collective between the seats. (01:56:15)
Revealing mistake: When the toilet fight is over, there's a pool of blood on the floor where Ethan steps on. However, he then moves around the place but leaves no footprints, revealing the blood is CGI.
Plot hole: When Ethan and Walker are waiting for Lane to drive by and knock him into the Seine, Walker realizes that Ethan was the agent who caught Lane. He says, "How long do you think he'll keep that to himself?" He says it as if Lane is going to see the White Widow or her brother, but since he hasn't left Ethan's side the entire time since the plane, wouldn't he know of Ethan's plan to take Lane before the brother could kill him?
Other mistake: When Solomon Lane is being extracted out of the sunken truck there are two divers breaking in. However just a moment later only Benji surfaces with the prisoner while Luther is on the lookout on the rib. (00:55:11 - 00:55:36)
Plot hole: Ilsa had no idea where the BMW was going - she was already on the bridge and in position when we see it turn left into the street, it could've easily have gone straight on.
Continuity mistake: When Ethan is chasing Walker up the brick tower, the hatch to the roof is in the middle of the tower when seen from the inside. However, in the exterior shot, we see that there is no hatch where Ethan is pretending to be exiting. Later we see the exit to the roof is on the side of the building.
Revealing mistake: The helicopter that picks up Cruise after the final fight is registered as if it is Norwegian, even though they're supposed to be in India. The scene was filmed in Norway.
Continuity mistake: The amount of blood on Hunt's face changes during the last fight against Walker.
Factual error: The number plate on the white van that Ethan, Luther, Benji and Elsa are travelling in has 8 digits instead of the 7 of a normal UK number plate.
Continuity mistake: In the bike chase, Ethan has side mirrors in all but one scene where he is surrounded by police coming from all corners. The bike's mirrors are missing then. (00:59:10)
Suggested correction: The number plate is the Norwegian format. Likely acquired whilst in mainland Europe.
The text on the side of the van says "London and the Home Counties" so driving on Norwegian plates would be incongruous and draw attention. Surely something you would not want in a covert van. If you're picking apart the van, why no phone number or website for the fake company. Again, it just looks wrong.
That doesn't change the fact that it has a valid number plate. Within the EU vehicles can be used in any country, provided they meet the local legal requirements. In the case of the UK, that is being up to date with any servicing and the vehicle being insured. It is not a movie mistake that the van exists or that it could be in London. They used a van they had access to.