When the leader of the soldiers is speaking, his mouth movements don't match what he's saying in Korean, and if you look closely, you can tell that he's actually speaking the English dialog given in the subtitles.
(at around 45 mins) As Ethan and Sarah are on the beach after Ethan gets her out of the Hospital, Sarah is dressed in hospital gown and slippers, she slips off her slippers and then walks down the beach. In the long shot she is wearing her slippers and then the next shot is a close up of both of them and in the lower left corner you see her slippers sitting on the beach where she took them off.
"Older" Ethan's eyes are brown, or perhaps dark green. In the scene where Ethan as a young boy is looking into the light from the magical chest, you can clearly see that his eyes are blue-green. This is most evident later in the same scene during the transformation from young Ethan to older Ethan.
The amount of Special Forces troops that enter the cave for the assault seem like a platoon. And yet, when they retreat after being confronted by the dragon,their number seems to have dwindled to a handful.
(at around 37 mins) A psychiatric facility would not put someone in a straight jacket for delusions or hallucinations. It's called "unreasonable restraint and seclusion". If the patient were a threat to his or her self or others, the person would be restrained.
(at around 57 mins) A character incorrectly refers to the US Bank building in downtown LA as the "Liberty" Tower, when in fact the Tower is actually known as the Library Tower. The LA Public Library is across the street and gave up the air rights for the Tower to be built (which helped to rebuild the Library after two devastating fires). This was possibly a deliberate error, as the building is occasionally incorrectly referred to as the Liberty Tower (most notably by President George W. Bush identifying it as a possible terrorist target). However, a local resident of Los Angeles would most likely not make this mistake.
(at around 1h 2 mins) Apaches are designed to be survivable against 23mm cannon fire. It is highly unlikely they would explode from the damage they suffered in the movie (crashing into a building and the street), especially not in a delayed manner (as the fuel tanks are armored.)
(at around 1h 3 mins) The soldiers controlling the .50 cal machine guns in the very front APC and tank are supposed to be Combat Vehicle Crewmen, yet they wear the uniforms and body armor of ordinary infantrymen.
In reality, firearms, especially those loaded with armor-piercing ammunition, would have easily penetrated the armor of medieval warriors, like that of the 'Artox' army, yet small-arms fire and even .50 caliber machine guns could not harm these warriors.
The cave assault force appear to be made of SWAT, but are delivered to the area in Army Black Hawk helicopters. Furthermore, the actual assault force would have been Military, likely Special Forces, with plenty of support, due to the distance from populated areas and the threat level.
(at around 1h 5 mins) During the city battle scene, there is a shot of S.W.A.T. team members firing on some flying dragons, who react with fireballs shot back at the officers. During this scene, you can see some of the officers running from the attack, in which a couple actually fly into the air well before the explosion appears on the rooftop.
In the lineup of military vehicles, the first two tanks are clearly mocked up British Centurions. However, the rest of the tanks are CG fakes, as evidenced by their lack of a raised lip on the glacis plate and the more correct shape of their hull fronts. They are also not obscured by the exhaust from the first two tanks.