In Rio, Blu narrowly escapes getting cut by a buzzsaw, which shears off a couple of feathers off the back of his head.
Happens to Sheeta at the end of Castle in the Sky when the antagonist, Colonel Muska, shoots off both of her braids.
During the Training Montage in Hercules, part of Herc's training involves throwing daggers at moving targets controlled by Pegasus. He misses most of them, and gives Pegasus a crew cut to go with it.
In The Man Called Flintstone when Bobo attempts to assassinate Fred thinking him to be super spy Rock Slag whom looks just like him, he throws a knife at his head it misses but splits his hair down the middle.
In Zootopia, a ram gets his belly wool shorn clean off when he lays flat against a tunnel wall to dodge a passing subway train.
Daigoro's topknot from Lone Wolf And Cub gets a chunk taken out of it during a fight, but he's otherwise unharmed. Considering that he's a small child, this might have been as far as the director wanted to go when it came to actually injuring him.
The protagonist receives one in a flashback at the start of the film but with an executioner's axe (one of his servants arrives just in the nick of time). "Thanks! I needed a trim!"
In the climax, the executioner is about to strike him again, but this time, the Baron's far-shooting servant shoots the axe out of the executioner's hands, and it spirals through the air slicing through the Sultan's headdress widthwise and giving him the trim; the Sultan has to keep wearing the ruined hat for the whole rest of the fight scene.
A variant of this happens early in Get Shorty, when Ray Bones tries to sneak into Chili Palmer's office to shoot him. Chili, having been alerted to this, just waits with his gun aimed at head height. When Bones breaks in, Chili shoots him — and leaves a groove on the top of his head from the passage of the bullet, to which Bones freaks out and goes running out of the building, screaming.
In Peter Jackson's depiction of the Battle of Dagorlad during the War of the Last Alliance in The Fellowship of the Ring, an arrow just barely misses taking Elrond's pointy ear off as it breezes through his hair.
The arrow was shot by a friendly elven archer standing behind Elrond, shooting at orcs charging at Elrond from the front. And his hair moving was probably from the arrow's turbulence, not the arrow itself.
In Robin Hood: Prince of Thieves Kevin Costner (Robin Hood) chops off a decent sized piece of Alan Rickman's (The Sheriff of Nottingham) hair during their final battle. He then quips "If I have to, I shall take you a piece at a time..."
The Thin Man in the Charlie's Angels movie cut off Alex's hair and sniffed it while swordfighting with her.
Red Cliff: At the climax, Cao Cao's topnot is undone (in Bullet Time, even!) by an arrow courtesy of Sun Quan.
Subverted in Kill Bill, Pt. 1, when for perhaps half a second, it appears that the Bride's katana has cut O-Ren's black hair off her head. Then you realize that O-Ren's scalp and the top of her skull have come off with it.
In many shorts of The Three Stooges, gun shots might leave a bald, steaming trail from the front to the back of Moe's or someone else's head, or else blow their hats (or toupees) off.
In Land of the Dead, Charlie fires at what appears to be Motown's head, only for the bullet to hit a zombie behind her while missing her head by a tiny margin. She tries the same trick later on, but the bullet grazes Charlie's head on the way to hitting the zombie.
Charlie: That wasn't close! You actually hit me!
In Crocodile Dundee 2, after Mick recruits a gang to help him rescue his wife from the Big Bad, one of the youths makes it clear that he's not going to participate, but instead plans to beat up Mick and take his jacket. Mick throws his knife, which bisects the kid's mohawk and sticks in the wall he's leaning against.
In Wanted, during Wesley's training with the Fraternity, he can't seem to bend bullets (don't ask) around a hanging body to hit the target. When Fox stands in front of it, Wesley is able to just curve the bullet enough to miss her, but clips a piece of her hair.
At the end of the "Parlay" scene in Pirates of the Caribbean: At World's End Hector Barbossa whips out his sword and slices off the dreadlock bearing Jack Sparrow's Piece of Eight in a single motion. It's unclear whether he would have cared much if he'd taken a piece of Jack's skull with it.