When MacGyver starts climbing the rope to rescue Zoe, the bottom is secured to something on the ground. When he has to hide from the guard, the rope is suddenly free to move about.
The first view of the holes in the rock shows the carved design clearly visible and in bright sunlight, but the next shot of the design is in shadow, with the holes partly covered/filled with dirt, and with the design positioned down lower to the surrounding ground.
When MacGyver sneaks onto the estate grounds and looks up at the flagpole, the sky is completely overcast/foggy. A few minutes later, Zoe looks at the flagpole from her cell window, and the sky is clear, with hardly any clouds.
Assuming the "optical pump" could actually work, it would depend not only on it being the right time of day, but the right time of the year for the sun to be in the right position.
When the black car pulls up in front of the house, its British style rear number plate can be seen to be loosely stuck on over an American style license plate recess. The number plate does not have the legally correct British lettering style or spacing, and also indicates a "U" year of issue - there was never a U year, T skipped straight to V.
When Mac and Zoe head back to their plane, they leave Zoe's archaeological kit behind. Skilled and caring "Earth advocates" like them would never leave items littering an excavation site, nor would they ever waste valuable scientific materials, especially since they were operating on such a tight budget. However, Zoe clan clearly be seen caring it with her on the way to the plane.
After the "optical pump" does its thing, Mac and Zoe walk in front of it, apparently oblivious to the fact that the next ray of sunshine could kill them. However, they were aware that the sun was behind a cloud at the time, and therefore there was no danger.
When MacGyver and Zoe arrive in London, they ask a taxi driver to take them to the Abbey of Saint Ambrose. But the abbey, long dissolved (possibly for centuries), is now a museum of torture implements (which will, of course, play a role in the plot). Aside from its being in London, MacGyver and Zoe do not know where the (former) abbey is located, so it is almost impossible the taxi driver would have known, unless he was part of the conspiracy, which he wasn't. The driver doesn't even ask for an address. Correction: This scene actually happens in part 1. Zoe doesn't give an exact address, but tells the cabbie to take them to the intersection of Whittingham and Bellows in Chelsea.
When Mac tosses the rope to the top of the wall, you can see the trajectory of his throw will cause the rope to hit the wall only about one floor up. When the camera switches to a long shot, you can see the rope traveled almost 4 stories to land on the roof - an impossibility given the low velocity it was thrown at, and the aforementioned low trajectory.
When Mac and Zoe find the Temple there are mountains in the background, which wouldn't be found in that part of France.
When MacGyver and Zoe are setting off from Weymouth to the airfield, the road is shown with a solid yellow line down the middle. No British roads have any yellow lines down the middle, it would be white in Britain, and a single line would not be solid either, but would be a broken white line.