When Mac is climbing up the tower in the power plant, the goon shoots at him and there are sparks/smoke coming from where the bullets hit. When the camera switches to a close up of MacGyver, there are no bullet holes in the wood or any indication that that area had just been sprayed with a machine gun.
The position of Lisa's body changes between shots after the lead bad guy kills her.
After the final fight with the bad guy on the beach, the back of MacGyver's shirt is completely covered with dirt/mud. After he goes back to the truck to stop the EMP cannon, his shirt is substantially less dirty.
When MacGyver is filling his Jeep with gas rain overflow is rushing past his Jeep. This indicates that there had been a recent rain storm, but there was no rain shown before this scene.
Modern vehicles have locking steering columns, yet Mac and Pete are able to drive the blue pickup truck without the key just by shorting the ignition wires. The steering wheel and gear shift would still be locked in place with the ignition key lock, even with its wires disconnected.
If Mac's jeep had run out of gas, he would not be able to just start it immediately after adding the gas and drive away as is shown here. He would either have to pump the accelerator and crank the engine a few times (carburetor) or click on the key and let the fuel pump run a few seconds (fuel injection) to get the new gas through the empty fuel line to reach the cylinders.
MacGyver looks at his watch, noting it was stopped (presumably by the EMP attack that took out his Jeep). His watch is a wind-up, so it would not have been affected by the EMP blast.
In the power plant when they are loading the super conducting magnet onto the trailer the max weight of the hoist is 5kg but the super conducting magnet weighs 14,500kg.
Smoke can be seen even before MacGyver detonates the grenade and blows up the cellar door.
Instead of rigging up the wire cables to destroy the EMP cannon, he could have simply unplugged it from the magnet.