Factual error: The boys go deer hunting somewhere in the Appalachians, but the geology is all wrong. That scene is in North Cascades National Park in Washington State. In addition, the "deer" they are hunting is a Red Stag. Not typically found in the Appalachians.
Factual error: When Michael Vronsky gets back from his tour he has a full grown beard. The only branch of the service that allows a beard is the Navy, and he was an Army Green Beret. The beard would only have been tolerated if Mike was either not in his Class A uniform or actively performing on a mission, neither of which are the case here. In addition, he's wearing a Black Beret, but his left shoulder patch is the 101st airborne, not Special Forces. Finally, he has six or seven hash marks on his right sleeve. Each hashtag is for 3 years of active duty service.
Continuity mistake: In the scene where the boys are changing into all their hunting gear on the side of the road, keep an eye on Christopher Walken. In a wide shot, he's pulling on a dark green sweater. There's a quick shot of De Niro, then back to the wide shot, in which Walken is again pulling on the dark green sweater.
Factual error: Back from Viet Nam, De Niro is wearing a green beret and he has a 101st Airborne Division patch on the left shoulder of his dress greens. It should be a Special Forces patch.
Factual error: When Mike is hunting after his return he switches from left hand shooting to right hand and his rifle also switches the bolt on the right to the bolt on the left.
Visible crew/equipment: In the beginning of the film, when Linda is getting ready for her wedding, standing in front of the mirror at home, a boom mic is visible at the top of the mirror. (00:07:45)
Continuity mistake: After Michael returns from Vietnam for the first time, he goes for a walk with Linda. As they start to pass in front of a store with a guy outside, Michael is shown fiddling with the collar of his coat. The shot changes, and though from their position and reaction of the guy outside it is supposed to be a direct continuation, Michael suddenly has his hands buried deep in the pockets of his coat.
Continuity mistake: When the car vs. lorry race occurs near the beginning of the film, as the car pulls in front the left wing of the bumper clips the lorry and gets noticeably misshapen. Then the shot changes, the car spins round, comes to a halt, and the damage has gone (and remains so for the rest of the film). Possibly made worse by the fact they then make a big thing of Nick actually getting out the car and inspecting it for damage in a disbelieving fashion that anyone could have pulled it off without damaging it.