MacGyver removes the lug nuts from the left rear wheel of Joey's Mercedes at the lumber yard, and later the wheel is shown with no lug nuts or studs holding it on. When Joey and his goons get in the car the lug nuts are back on, but when they drive away the wheel falls off.
MacGyver cuts a radiator hose on the US Marshal's car and antifreeze gushes out. When the Marshals leave with Frank there is no puddle of antifreeze where the car was parked.
The tow truck's winch reels out as the car is lifted; to raise the car, the cable would have to be reeled in, not out.
MacGyver is seen picking a lock with a Swiss Army Knife. This cannot be done with the tumbler lock mechanism as shown on the knob.
When MacGyver uses the zip-line to escape from the mansion, a couple of shots clearly show the stunt double (Walter Robles) instead of Richard Dean Anderson.
The tow truck is not hooked up to the car before Mac lifts it.
When Frank is kidnapped, before the car skids to avoid the garbage truck, the skid marks from a prior rehearsal are visible.
Mac is a deeply-dedicated environmentalist, so he would not throw away his junk mail; he would put it in a paper-recycling bin.