After the minis cross the weir, a British number plate is visible even though they started the chase with Hungarian number plates. This is because most of the chase footage is from The Italian Job.
While MacGyver is riding the recovered horse and escaping from the bandits, there is only a chest strap around the horse from the saddle and a neck strap. But when the helicopter picks them up there is also a belly strap and a padded strap behind the rear legs.
The car in which Hungarian Inspector Jan Messic and Russian agent Nikolaj Kossov chases MacGyver and the gypsies in the 3 Minis changes from a white Lada to a brownish Alfa Romeo between the footage used from The Italian Job (1969) and the actual MacGyver footage.
When MacGyver exits the tent after stealing back the horse, the sky is clear, bright blue - no clouds. Just a few moments later when he reaches the beach, the sky is completely overcast and gray.
The prison camp is supposed to be somewhere in Hungary, but it's located in a somewhat arid desert-like place, and since Hungary is a landlocked country with no beaches or arid places, it belies the filming location in Southern California.
When MacGyver is sliding down the roof, the stunt-double can be seen.
MacGyver is in Budapest, Hungary, to meet a contact. This contact is a Hungarian, but the newspaper he is reading is called "De Telegraaf". This is actually a Dutch newspaper.
Mannequins are visible inside the car crashing through the sewer grate towards the end of the chase scene.
Though this is supposed to be in Hungary, Italian landmarks can be seen during the car chase.
Budapest is located in the Eastern part of Hungaria. The border to Austria is several hours away. Therefore, the car chase cannot possibly reach the Austrian border after just a few miles driving.
When MacGyver catches Jana in the rubbish bin, she says, "Okay, MacGyver, I'll get your watch." At this point, MacGyver hasn't given his name to her; she gives her name to MacGyver the first time he catches her stealing his Swiss army knife, but he does not return the courtesy.