Picture of Harry Truman in the Sheriff's office, typically the President's portrait in a law enforcement office would be the current office holder. At the time this film was made (1962), it should have been a portrait of JFK.
The helicopter pilot reported that Burns fired at his tail rotor (to allow him to land safely). However, without actually seeing the trajectory of the bullet, angle of the rifle, or the impact (it missed), the pilot had absolutely no way of knowing Burns' intentions.
When the Air Force Bell helicopter first arrives, the registration number N8441E is clearly seen as it departs from the hover. Later, when you see the close-up of the tail rotor as Burns is shooting at the helicopter, it is clearly marked N8411E. They apparently, used two similar helicopters with similar registration numbers whilst filming the movie.
When Jack Burns (Kirk Douglas) is about to break out of jail, he didn't have his hat after coming back from the beating by Gutierrez (George Kennedy). However, when he's on the other side of the jail cell saying goodbye to Paul (Michael Kane), he reaches behind him and puts his cowboy hat on.
A common Hollywood mistake when radios are used is to end the transmission with the phrase "Over and out". "Over" is used when turning the channel to the other speaker and "Out" is used when the transmission is finished, but one wouldn't say "Over and out". In this movie, radio use is inconsistent. Sometimes "Over and out" is used incorrectly, sometimes "Out" is used correctly and sometimes speakers switch back and forth without indicating anything at all.
During the scene when the Sheriff is facing the hovering helicopter and talking to the pilot by radio, on Matthau's closeups, the shadow of the helicopter has been positioned behind him. Only the sun is behind the Sheriff and the helicopter's shadow would be way in front of the jeep, not behind.
The aircraft purports to be a US Air Force helicopter and indeed shows some military markings. However, the FAA "N" number (N8411E) is clearly visible on the belly and the tail of the aircraft. Military aircraft never show civilian markings.
When the one-armed man throws his mug of beer at Burns and it hits him in the thigh, it shatters. No heavy glass mug of the type used in bars would shatter (or even break for that matter) when hitting something that soft.
At the 1:33 mark, whilst Burns is resting with his horse, Whisky, under a tree, you can see a thin black line attached to the horse's bit, coming in from the left. It appears that it is being pulled on by a crew member to keep Whisky's head up from where she is grazing.
At the end of the movie, while the camera is focused on Kirk Douglas' face, he is making significant starting actions with his eyes, conveying the character's thoughts. It happens to be raining very hard through this scene, yet his expressions are unimpeded, revealing that the rain is being simulated between the camera and him.