Revealing mistake: During the bathroom fight and Ethan is starting to stand up from a urinal, you see his knee go into the foam padding that was used in the room.
Revealing mistake: At the end as Hunt and Walker are fighting for the detonator on the spectacular cliff (in actual Norway) they fall off, grabbing onto the rope suspending the helicopter. The issue being the rope moves too easily, revealing it was slack.
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: The fight scene in the nightclub toilet between Hunt, Walker and who we believe is Lark is brutal. Both Hunt and Walker take several blows to the head and face, but when it is over there isn't a mark on either of them.
Continuity mistake: At the start of the truck chase the road is completely dry and it's sunny, as they go over the bridge the road is completely wet and it's overcast, then the road is dry and it's sunny again.
Continuity mistake: When the truck gets knocked into the river it is almost submerged by the time Ethan reverses the truck. Cut to part of the truck chase and then back to the river - the truck is sinking again and the driver and passenger are getting out. The truck is also at a different angle to when it first went in.
Continuity mistake: During the BMW/Bike chase in Paris, Ethan crashes through some parked motorbikes and drives away, the rear window on the passenger side is completely open. Before and after this scene the window is shut.
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 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.
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.
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.