Loki has an awesome one near the beginning of Thor when Thor decides to journey to Jotunnheimr.
In Hellboy II: The Golden Army, a flashback detailing the backstory of the title golden army has a group of pseudo-stop motion wooden puppets acting it out. Whenever the elf king is described as being upset, he declines into a facepalm.
In The History Boys, Timms performs an admirable Head-Desk in exasperation with his schoolwork.
In the RoboCop series, the President of OCP does this at least once in every movie. He does it after ED-209 malfunctions in RoboCop, and again in RoboCop 2 after viewing the suicidal RoboCop prototypes.
Natural Born Killers: Scagnetti does this over his own stupidity when Mickey says, "I was out of shells." Scagnetti combines the Face Palm with "D'oh!" as much as that fork in his throat will allow.
Arthur: Who leaps out? Bedevere: Well... Galahad, Lancelot, and I... Leap out of the rabbit. Arthur and Lancelot: (facepalm)
Annie Hall: Alvy Singer does this in a flashback to his childhood:
Alvy Singer: (voice-over) And Ivan Ackerman. Always the wrong answer. Always. Ivan Ackerman: Seven and three is nine. Alvy Singer: (facepalm)
The 1939 Screwball ComedyIts A Wonderful Worldnote Yes, "World", not "Life". features back-to-back scenes with facepalms, first from Jimmy Stewart's wrongly-accused fugitive, then from the police chief leading the manhunt for him; both are reacting to disastrous "help" they are getting from other characters.
In Groundhog Day, Phil does a head-pillow variant when he tells a shrink his problem and the shrink answers, "Can you come back tomorrow?"
In Back to the Future, Doc Brown does this right after Lorraine asks Marty to the dance, suddenly realizing that Lorraine (Marty's mother) has developed romantic feelings for Marty.
Flight Director Gene Kranz reacts with a subtle one and some exasperated snarking on learning that the only available spare carbon-dioxide scrubbers on the stricken spacecraft (from the dead Command Module) are square, and the receptacle for the only working scrubber system (in the Lunar Module) is round.
Gene Kranz: (facepalm) Tell me this isn't a government operation... I suggest you gentlemen invent a way to put a square peg in a round hole. Rapidly.
Another one happens a little later on, when Mission Control MacGyvers a solution, which includes using their spare urine bag. Which leads to this exchange:
Fred Haise: Shit, I tore it. Jack Swigert: Shit. Fred Haise: Houston, what do we do if we rip the bag? Can we tape it? Andy (CAPCOM - WHITE): They just tore the bag. Technician: (facepalming) Oh, no.
Nathan Lane's character Max Bialystock in The Producers has a record number of these throughout the movie.
Beta does one when he sees a Zandozan ship land in the distance in The Last Starfighter. It's a rather unusual example, as his head isn't attached at the time.
The Museum of Fine Arts Boston is advertising the Turkish film Last Stop: Salvation with a screencap featuring a Face Palm.