When Cory is looking through the pictures in Natalie's room near the end of the film, he sees a picture of Natalie and Emily in graduation gowns. Yet, Cory had said earlier in the film that Emily had died three years earlier at the age of sixteen. Natalie doesn't look substantially younger in this photo than she did in other parts of the film, so this would appear to be a high school graduation photo for a high school graduation Emily didn't live to see.
The Carhartt logo on Cory's Jacket disappears and reappears. For example at around 10:00 minutes when he looks at the tracks of the snow lion it's there but when he gets off the snow sled a moment later it's suddenly gone.
When Cory is talking to Martin about Chip, Cory is tight-lipped and on the verge of tears at one moment when Martin asks if he's stay with him. The next shot shows a much more stolid serious face and a slightly open mouth on Cory.
In the real world, if the oil drilling contractors drew their weapons on federal officers (FBI and BIA) and county sheriff's deputies, the oil field workers would have been placed under arrest, immediately.
The medical examiner states he cannot call the death a homicide, and the FBI agent says that hinders her investigation. While he correctly states the *cause* of death, there are five options for *manner* of death: natural, accident, homicide, suicide, and undetermined. Until the death and its circumstances are investigated and known, he would most probably (if thorough) have called the manner of death undetermined for the time being.
It's not possible to freeze the lungs by running, even in extreme cold. Air gets heated by the nose and throat before it reaches the lungs, and the lungs themselves are continually heated by the circulating blood. You'd die from hypothermia long before the lungs could freeze.
The US Fish and Wildlife Service, Wildlife Officers are usually uniformed officers, with full peace officer status, and do not have the duty of trapping and controlling predators. Take it from a retired USFWS Wildlife Officer who has had the undesirable duty of assisting the FBI with a murder investigation on the Ute Native American Indian reservation in Utah adjacent to the Ouray National Wildlife Refuge. The USFWS truck and decals were correct, but the duties carried out by Jeremy Renner in the film are those of the US Department of Agriculture, Wildlife Services Division (non-Peace Officers).
The goof items below may give away important plot points.
The reason Jane cannot request the FBI team she desperately needs at the reservation is because Natalie's demise cannot be labeled a homicide on her death certificate despite the clear circumstantial indications of foul play. Even if Jane's shootout with Sam Littlefeather shortly thereafter didn't incur the expanded involvement of the FBI, the discovery of Matt Rayburn's body almost certainly could; even an amateur medical examiner would've been able to determine his cause of death regardless of his state of decomposition from scavengers, and it would be fairly easy to establish that the previous victim's boyfriend did not fall off a sledge naked in the middle of the reservation.
When Jane and Ben go to question Sam Littlefeather, Cory spots sledge tracks headed up a nearby ridge with no tracks coming back, which ultimately leads to the discovery of a second body and the revelation that the culprits are from the oil drilling site. This means that members of the security team would have had to travel the 50 mile refinery road to get to reservation in a vehicle heavy enough to carry a motorized sledge, which they then unloaded in that specific spot so its driver could hide a corpse. Why go through the trouble to do that if you're just going to leave a trail leading back to the scene of the crime?