When Duke and Samantha are talking while she's examining the creature (right before he goes to get the power bone saw), he corrects her by saying the Marine term is 'Hooah,' when in fact it's 'Ooh rah'. 'Hooah' is an Army term, not a Marine term.
Reaper repeatedly addresses Sarge as "Sir." Sarge, like all the marines, is an enlisted man. Enlisted personnel are NEVER addressed as "Sir" in any branch of the U.S. armed forces (and most Western militaries, also). Also, at least once he refers to himself as the squad's "commanding officer," which is a specific term referring to an officer in command of a much larger entity, usually no smaller than a battalion, squadron, or ship. (This concept is rarely used correctly in movies and on TV; it's often incorrectly employed as a military-sounding term for anyone who is senior to another military member.) As the senior marine in the squad, his title would be "squad leader."
When Reaper, Sarge and Sam are watching the video file from UAC's human subject test, the time-code on the screen first starts out at 2:23. After a quick shot of all 3 actors, the timer then reads 2:33, then 2:34. It then instantly switches to 3:24 without any apparent fast forwarding of the video. After another quick shot of Sarge, the timer then starts at 2:27.
When Duke and Destroyer hear something in the vent above them, bullet holes can be seen in the duct to the right of the grate. The holes disappear when Destroyer moves to open the grate and reappear after Duke and Destroyer shoot the duct full of holes.
When Sarge asks Reaper how much ammo he has left, Reaper replies that he has "half a clip." The correct term is magazine (usually abbreviated to "mag.") Any Marine would know the difference between a clip and magazine.
Although this may be the case in reality, the Doom games refer to small quantities of bullets as clips; (e.g. 'picked up a clip') calling it a clip in the movie was probably intentional in order to be more consistent with the game.
When Portman and Goat check the chemical levels in the airlock, Portman mentions magnesium, chromium, and lead. We see his "chemical detection" device, and in the order he stated, follow the symbols Mg, Cr, and Ld. The symbol for lead is not Ld, but Pb. However, this is not necessarily a goof, since the device is not labeled as a Periodic Table, and the display may be "dumbed down" for common usage by lay people. "Ld" is a reasonable abbreviation for Lead for people without a chemistry or Latin background (Latin for metallic lead: Plumbum).
No Marine would ever refer to a sergeant of any rank as simply "Sarge." This is only done in the Army and the Air Force. However, this could be because it is the Sergeant's "battlefield nickname", considering several other members of the squad are also called by nicknames.
The hand held video game Duke is playing (Galaxian 2 by Entex) is meant to be played holding the game vertically, one side of the buttons control the ship, the other side the Galaxians. Duke is holding the game horizontally, playing both sides at once, while seeing the action sideways.
The goof items below may give away important plot points.
In the bathroom stall just before Portman dies, the screen shows what he is seeing. The view of the gun on the right side of the screen, suggesting it is being held in a right handed manner, Portman however holds his weapon left handed.
When the Monster grabs Pinky, Reaper and Sarge go after it. When they round the corner, they fire their weapons through a Nano-Wall at the monster. However, in the next scene, the Nano-Wall disappears, then re-appears when the horde of zombies arrive.