Factual error: When Chigurh is in the gas station and throws the wrapper on the counter, the "Nutrition Facts" are visible on the label. Set in 1980, the "Nutrition Facts" label wouldn't have been available for at least another fourteen years, as nutrition labels were started in 1994.
Character mistake: When Moss is arguing with the border guard at the Eagle Pass international bridge, he claims that he is a veteran of the "12th Infantry Battalion." There has never been such a thing as the 12th Infantry Battalion in either the Army or the Marines. Rather, they are based on a structure of 3-4 battalions per numbered regiment (i.e., 1st Battalion, 3rd Infantry Regiment/2nd Battalion, 5th Marine Regiment, etc). The film takes this seriously, as the guard, a veteran himself, buys Moss' story.
Suggested correction: He might have meant 12th Infantry Regiment. From Wikipedia: "Three 12th Infantry battalions deployed to South Vietnam with the 4th Division from August through October 1966."
If he said "battalion" but meant "regiment", then it's still a valid mistake for saying it wrong and being believed.
Regiments have not existed as functional units in the US Army since shortly after Korea; they are simply historic names associated with various battalions. Marine battalions are not numbered higher than 4 in any regiment, and in any case do not carry an explicit designation of "infantry."
Visible crew/equipment: When Moss is talking with the security guard at the border bridge, you can see the camera and the cameraman next to Moss in the reflection of his sunglasses.
Audio problem: When Anton Chigurh challenges the gas station proprietor to a coin toss, at a certain point he says "It's either heads or tails." If you listen carefully it's clear that the phrase said by him has been dubbed.
Audio problem: When, at the end of the movie, Chigurh leaves the accident site after he said to one of the boys: "You didn't see me, I was already gone", the boy answers saying: "Yessir", but his lips don't move.
Suggested correction: The boy's mouth is open. You can say "Yessir" with your mouth open without moving your lips. Try it.
I can't.
Factual error: In the gas station, the snickers logo is the one they started using in 2005.
Continuity mistake: When Llewellyn places the money satchel into the motel (room #138) air vent there are a number of dust scratches to the bottom left of the vent. Then later on when Chigurh opens the vent with his dime and you look at the character shot from the inside the vent (from room #38), the scratches to the left are not there.
Other mistake: It would have taken two hundred $10,000 wrapped stacks of $100 bills to equal $2 million. There is no possible way that suitcase could hold two hundred of them.
Factual error: The satchel isn't large enough to hold $2 million. Maybe $1 million. There were six stacks showing on top, so $60,000 across the top layer. Each stack is nearly an inch deep, meaning that the satchel would have to have been about 30 inches deep, in other words 2 and 1/2 feet deep.
Visible crew/equipment: A crew member seen standing on the far left of the frame when Llewelyn is being chased down the river. (00:19:34)
Continuity mistake: During the dialogue between Moss and Wells at the hospital, there is an ashtray on the bedside table next to him changing position in various shots.
Continuity mistake: Near the end of the movie in the car accident scene, when Anton steps out of the car completely injured, the door's window appears broken in a way and then in a different one in a next scene.
Continuity mistake: After Chigurh kills the three Mexicans inside the motel room, the dead body of the second man with the meter in the hand changes position in two detached shots. At first he has his head on the ground, then he keeps it pressed from behind against the cistern toilet. (00:50:28 - 00:51:09)
Continuity mistake: When Moss unloads the shotgun onto the diving Chigurh, Chigurh's right thigh is towards Moss, but later when C fixes his wound, it is his left thigh he's repairing. (01:04:14)
Factual error: When Josh Brolin is leaning on the cinder block fence at the motel towards the end of the film, behind him are some mountains and a tall building. The scene is set in El Paso, Texas but the mountains shown are the Sandia Mountains which are east of Albuquerque, New Mexico.
Factual error: The phone bill shows several calls from Area Code 210 (San Antonio). This Area Code did not exist until 1992, some 12 years after the movie is set.