When Spencer gets shot, the gun is pointed down and to her left, but when she falls to the floor she grabs her middle-right torso, nowhere near where the gun was pointing when it went off.
At 18:57, Colson puts a lit cigarette in his mouth, but in the next shot, the cigarette is gone.
When MacGyver is making a sort of bomb, he opens a metal tin with the sodium in it. Sodium cannot be stored in anything metal, not unless you want an explosion to take place. Ideally, it should be kept in a glass jar with a screw top, submerged in paraffin. That glass jar should then be kept in a plastic container. Only small amounts should be kept in any one container. A group of metallurgy scientists would not be storing pure sodium metal so haphazardly.
Pure sodium is extremely reactive to oxygen. As soon as MacGyver opened the canister it should have begun reacting. What's more, it would have been stored covered in some kind of viscous oil, in order to keep any oxygen from reaching it and setting it off. In addition, the few grams that Mac used would not have been enough to have caused that big of an explosion.
Mac mistakenly says that scorpions sting themselves to death. Scorpions do not actually sting themselves, and even if they did, they are immune to their own venom.
MacGyver shatters his binoculars to obtain a lens to reflect the laser back on itself. Mirrors reflect, not lenses.
Breathing the extremely corrosive acid fumes coming from the damaged tank would have burned the mucous membranes and lungs of MacGyver and Spencer, yet they show no ill effects at all.
The hole that MacGyver has blasted in the wall and then climbs through changes shape as he enters and exits on the other side.