Done by The Carpet, of all characters, in Aladdin; especially impressive, as it doesn't have a face or a palm. Immediately followed by Aladdin himself, as he's trying to explain to Jasmine why he lied to her about the first time they met. Even Genie does it in the same scene. Well he hits his head, but still, it counts.
Towards the end of Disney's adaptation of Alice in Wonderland, during Alice's trial, unbirthdays are brought up, which causes a whole new party, and of course the Mad Hatter and the March Hare have to sing "A Very Merry Unbirthday to You" for the Queen of Hearts.
Before they meet Anya, Dimitri and Vlad are holding an audition for girls to play Anastasia. One prospect is a middle-aged woman, who says in a sultry voice, "Granmama! It's me, Ana-STASIA!" Their reaction is a combination Head Desk and Face Palm.
He's particularly prone to the "depressed" type doing it in the prologue, (wrapping both arms completely around his head in an effort to hide himself), and a few times in the mid-quel Beauty and the Beast: The Enchanted Christmas.
Hiro adopts this pose at the end of the first trailer for Big Hero 6 after his attempt to suit up Baymax ends in failure.
In Bolt, one of the mooks gets zapped when trying to facepalm while wearing an electric glove.
Brave has Merida pulling her wimple over her eyes when Lord Macintosh presents his son as one of her suitors, clearly disgusted by his macho show-off routine and the situation in general. She facepalms directly when the brawl starts for the second time and her father joins the fray.
The coach of the baseball team the title character is on does this when Chicken Little swings the bat when he was just told not to.
Kirby also does this after trying to explain to Buck Cluck multiple times where his parents are, of course it doesn't help that all he speaks is gibberish.
After when She says "Oh, Ace, NOOO!!!". Ace Does That.
Despicable Me has Gru pinching the bridge of his nose upon seeing Dr. Nefario's "boogie robots". Gru holds up his palm when Nefario is about to say something and then quietly tells him that he asked for "cookie robots". He then walks away wondering why Nefario is so old.
In Finding Nemo, Crush does this when Marlin says he needs to get to the East Australian Current. "You're ridin' it, dude!"
Kristoff does one in Frozen when the trolls tell him in the song "Fixer Upper" that he should get Anna's "fiancé out of the way" so that he could be together with her.
The Great Mouse Detective. Basil has a particularly pained one when Dawson gets drunk and makes a fool of himself onstage when they're supposed to be undercover.
Done in Kung Fu Panda 2 whenever Po makes an idiot of himself. A notable version is Crane who, lacking hands, pulls his hat down over his eyes instead (it's the scene where Po destroys the toy cannon thinking it's the BFG).
In Lilo & Stitch, after Stitch escapes from his confinement and goes to hyperdrive, Captain Gantu does this while saying, "Get me Galactic Control..."
In the first film, Beth groans and facepalms herself while in Gordy's police station to confront Shaw.
In Open Season 3, Gisela and Giselita do this sometimes (and at the end, they do a two-handed facepalm together as the ladies are starting to dance with Ian, who is their coach). Stanley also does this after Roger says, "It's destiny!" when the pets come to save Boog from the Maslova Family Circus.
Dr. Faciller pinches his nose after his gang of evil shadows takes a severe beating from a firefly.
Maleficent in Walt Disney's Sleeping Beauty does this when she realizes her minions have been looking for a baby... for 16 years.
Maleficent: Oh, they're hopeless. A disgrace to the forces of evil.
In Shark Tale, Don Lino does this and says, "Oy vey." in annoyance at his henchman Luca putting the record on the wrong song, which was "Baby Got Back (I Like Big Butts)" by Sir Mix-a-Lot. Frankie also does this.
Strange Magic: The Bog King does this when his bodyguards fall in sleep in front of him.
A mass one occurs in The Swan Princess. Princess Odette and Prince Derek are expected to go through an Arranged Marriage but because the idea was formed while they were kids they started to hate one another after being forced to spend multiple summers with another. However, upon seeing one another as adults they instantly fall in love and Derek announces to start the wedding. Odette isn't too keen on this and asks Derek why the rush. Derek responds it's because he thinks Odette's beautiful. Odette is flattered by the compliment, but wants to know if that's the only reason Derek wants to marry her. Derek responds "what else is there?" His mother promptly started crying and everyone else groaned at Derek's sheer lack of tact.
Lord Rogers: You should write a book. How to Offend Women in Five Syllables or Less.
In Tarzan II, Kago does this gesture after his brother Uto says that if their bones were hollow and they were able to fly, they'd be the "biggest birds in town".
Lord Victor Quartermaine: How on earth would those tiny-minded buffoons ever catch such a big rabbit? Lady Tottington: Mr. Wallace? Wallace: Um... with a big trap! Gromit:[facepalm])
There's also Victor doing this after PC Mackintosh says "the beast [were-rabbit] isn't actually dead yet?!" with a megaphone. It's even funnier while said character is doing the gesture, the said officer said "oops" with a megaphone.
One is shown in Wreck-It Ralph when Vanellope learns how to drive. Ralph is trying to keep a safe distance from her so her car doesn't hit him (again). When Vanellope's car hits a stalagmite, Ralph does a facepalm as it falls down on him.
Paul does this in Yellow Submarine after Ringo cracks a pun regarding the sub's broken engine:
John: Maybe we should call a road service. Paul: Can't, no road! Ringo: And we're not sub... scribers. John, Paul and George: "Sub scribers"... ooh. [Paul facepalms]