Deliberate mistake: When the tower is looking around with the binoculars, the picture shows night vision. The binoculars were regular. (00:10:30)
Other mistake: Near the start, instead of showing bullet casings drop they show full bullets drop from the gun on the ground. (00:12:44)
Suggested correction: The shell casings appear to be unfired, but this is only because of the lighting and angle. Upon closer examination they are empty 5.56mm brass as they should be.
Continuity mistake: Spoiler alert! At 20:54 Nile gets her throat sliced and it's on her left side, when she wakes up the bandage is in the centre of her throat. (00:20:54 - 00:22:35)
Suggested correction: You can see the majority of the cut is in the center of her throat. When her head was turned, her neck wasn't, so he didn't get much of the left side of her neck, but the bandage wraps around enough to cover the left side of her neck that did get cut.
Plot hole: The two immortals get captured, they take out all the guards inside the transport van, but they don't break free of their thin zip-tie's which aren't even handcuffs, and they don't fight the ground team waiting for their arrival. It doesn't make sense why they board the plane and don't fight Copley and his team of mercenaries or soldiers. They survive gunshots, and all that's pointed at them is traditional firearms. So the logic here? (00:59:34 - 01:01:00)
Suggested correction: Taking out unsuspecting guards in a confined space with the element of surprise is one thing. Taking out guards who are primed for their arrival and outside melee range is another. As for the zip-ties, the immortals aren't super strong or anything. They probably have no way to break or cut the ties.
Continuity mistake: When driving the silver Audi A6 the first time, the version they are driving is a diesel, seen by the exhaust pipes facing down towards the ground with a bend. In the next shot they are driving the petrol version of the same A6, seen by the exhaust pipes holes facing directly out from the car. (01:18:33)
Deliberate mistake: The immortals retrieve firearms from the truck and arm themselves, however they do not take any spare magazines. A handgun's average magazine capacity is 15 +1 rounds, that can be emptied in 2 seconds in rapid fire, Jerry Miculek fired 27 rounds from a 1911 in 3.5 seconds; he's a world record holder but still this gives some perspective on how fast ammunition depletes. (01:18:45 - 01:19:20)
Visible crew/equipment: After the CEO examines the ax just as Andy goes through the double doors after being cut free in the lab a crewmember pops his head up from the floor, then the camera angle changes. (01:37:00)
Plot hole: Nile checks the gun Andy gave her and realises it's empty, which in turn leads her to realise that Booker was setting them up, because he gave that gun to Andy in the first place. No way that Nile, a marine, and Andy, a timeless warrior, somehow both missed the noticeable difference in weight between a fully loaded pistol and an empty one.
Suggested correction: That is only true if you handle the same gun all the time. Throughout the movie they shoot dozens of different guns, all with their own loaded/unloaded weights. Different guns are also made of different materials, and mixture of materials, which would change the weight. Different guns are also balanced differently, depending on the materials and manufacturer. The weight difference in a lot of the guns they were carrying between loaded and unloaded were between 3.5oz-7oz (which is not that much). With all the different guns they use and carry throughout the film, it is not a mistake that they wouldn't catch it. Also, Nile is used to carrying an M16 or M4, not the handgun used in the film so she would have no way to know the loaded vs unloaded weight. They would also not expect someone in their team to betray them, so there's no need to check the weapons (although you should check any gun that is handed to you).
Other mistake: All their wounds heal immediately and preventing wounds from closing is painful, but Andy has earrings the entire time and Nial has earrings in the final scene.
Continuity mistake: After the initial attack on the church we seek camera shots of the interior as if Copley has access to CCTV images of the interior (rather than helmet cams), but Copley does not see anything that happens in the room again. The last soldier killed at the church gives a signal to the soldier behind him, but there is nobody following him.