Revealing mistake: When Swagger is on the mountain and looking though his scope at the guy with the shotgun, the audience is shown what Swagger sees though his scope. However, the pictures we see does not match that very telescopic view Swagger would see at that distance but instead something shot with a relatively wide angle lens at close distance.
Factual error: When FBI agent Memphis is reviewing the assassination on video, you can hear the shot first and then see the blood spray of the bullet impact on the Archbishop. This defies physics as bullets travel faster than the speed of sound. Given the distance the sniper shot from, the sound of the shot would be be heard a second or 2 after the impact.
Continuity mistake: When Memphis is taken hostage his telephone falls on the middle of the sidewalk. In the next shot it lies on the edge.
Revealing mistake: When Gunny snipers the shotgun (and hand) off the bad guy in the snow covered mountains, his rifle is right-handed bolt, and he shoots right-handed except for the mountain scenes. He closes his right eye before he shoots, and it even shows a close-up of his left finger on the trigger, obviously a flipped shot.
Other mistake: In a whole lot of scenes, Bob closes both his eyes when he fires his sniper rifle. This is a common and natural reaction when an inexperienced shooter fires a gun. An experienced hunter or marksman would never do that, and especially not a military sniper.
Audio problem: Right after Swagger takes the first shot at the helicopter with his M82A1, we see him from the behind reaching his hand to reload the rifle. We hear this metallic sound of reloading but if you look carefully, his hand doesn't even touches the cocking handle.