Both Scrooge himself and Gladstone Gander wear spats, though without shoes, to signify that they're a little "finer" than most of the other male Ducks (Donald, Huey, Dewey, Louie, Fethry and others), who go around barefoot.
At one point at the Alvin and the Chipmunks sequel, Eleanor wears platform sandals because Ian insists she has to be taller. Once the Chipettes realize Ian was Evil All Along, Eleanor takes off the shoes and throws them at his face.
Tanya in An American Tail 2: Fievel Goes West picks up a pair of slippers as part of her costume when Cat R. Waul recruits her as a singer for his new saloon.
And in the first movie, Bridget wears shoes too. (This has a related continuity error; in some scenes she's barefoot like the other mice.)
As well as Reed Dailey in the fourth movie, An American Tail: The Mystery of the Night Monster.
Another Don Bluth example would be Patou the dog from Rock-A-Doodle. The reason why he had to wear shoes is because he is constantly suffering from bunions, and to him, his shoes make his feet comfortable.
Chanticleer the rooster is normally barefoot whenever he's still living on the farm, but later gains shoes after moving to the city and becoming a famous Elvis Presley-esque musician.
Cousin Kristofferson in Fantastic Mr. Fox is the only animal character in the entire movie who regularly wears shoes.
Beaver's son also wears shoes as a part of his Whackbat uniform.
Most of the characters in the Geronimo Stilton series do not wear shoes, despite wearing every other article of clothing imaginable. One of the few recurring characters to wear shoes is Geronimo's assistant, Pinky Pick, as part of her young-and-crazy style (they're platform shoes with fish swimming in them).
In the first Redwall novel, it's often made note of in the early chapters how Matthias wears sandals that are too big for him (causing him to trip). Later Redwall villains of pirate/corsair variety occasionally have boots, and of course, there's Captain Tramun Clogg, famous for... well, guess what kind of shoes he wears.
Although not anthropomorphs, Hobbits from The Lord of the Rings are known for not wearing shoes. In an early draft of LOTR, the character that eventually became Aragorn was a badass adventurer hobbit who was The One Who Wears Shoes as his feet had been damaged through torture in Mordor.
The turtle Franklin in the book and TV series by the same name... sort of. Most of the stories open with the phrase "Franklin could count by twos and tie his shoes," but Franklin actually isn't depicted wearing shoes most of the time. However, Franklin is the only character shown wearing shoes in the series. Also, most of the film Franklin and the Green Knight is set in winter, but Franklin is the only character seen wearing boots. All of the others are barefoot, or barepaw, as the case may be. Some of the later books do depict his sister Harriet wearing shoes and if Franklin is wearing boots, it's a safe bet she'll be wearing them too.
Richard Scarry's animal characters are generally shoeless, except for Lowly Worm, the one member of the cast who doesn't have feet.
Gonnigan from the Sesame Street segment "Abby's Flying Fairy School."
Most of the Fraggles of Fraggle Rock go barefoot (at least when we see their feet), but Traveling Matt very clearly wears both socks and shoes — maybe because he's the one character who spends the most time walking around in the human world.
In the first two games of the Donkey Kong Country series, Funky Kong wears sandals, the only Kong in either game to wear any sort of footwear. In the third game he wears boots, but Wrinkly gains sneakers.
Swanky Kong also gains black shoes with spats in DKC 3 (though not in the GBA version, where he's dressed the same as in DKC 2).
Also, among the Animal Buddies, only Expresso the Ostrich (DKC 1) and Squitter the Spider (DKC 2 & 3) wear sneakers. These also happen to be the only land-based Animal Buddies who lack a Goomba Stomp...
The cartoon series also had Candy Kong wear sandals.
Tiny Kong wore white shoes in 64, and sandals after the characters were redesigned by Paon. In fact, she's the only one who still wears any kinds of shoes after the redesign.
Sonic the Hedgehog: Chip in Sonic Unleashed is the inverse of this: he is one of the few characters who does not wear shoes. Given how running really fast is practically mandatory for all Sonic characters, many of the characters wear shoes and nothing else.
Breezly Bruin from The Peter Potamus Show. Also the Goofy Guards Yippee, Yappee and Yahooey.
SpongeBob SquarePants, though stretching the definition of Funny Animal as far as it can go, has titular character himself be one of the few characters in the show to wear shoes. Sandy is also another one, but they come with the diving suit. In her home, she's a straight-up Half-Dressed Cartoon Animal.
Lampshaded in the episode "Your Shoe's Untied," where SpongeBob forgets how to tie his shoes and can't find anyone who can help him. He finally has his shoes tied by Gary, who is secretly wearing sneakers under his shell.
Lampshaded again in the episode "Chocolate With Nuts." Spongebob and Patrick are looking through a magazine about fancy living, and after looking at one picture, Patrick comments "And this guy's got shoes!"
Archaeologist Katie Dodd in the TaleSpin two-parter "For Whom the Bell Klangs" is one of the only characters to wear shoes on the show, period. Rebecca's dress while she's under the ghost captain's spell in "Her Chance to Dream" seems to include shoes as well, though this is inconsistent from scene to scene.
Also, within the main cast, Don Karnage and Gibber are the only recurring characters who regularly use footwear.
In the children's cartoon Timothy Goes to School (based on a children's book series called Yoko & Friends), the fox characters Lilly and Mrs. Jenkins are the only two characters to wear shoes.
On Tiny Toon Adventures, most of the non-human characters don't wear shoes. The ones that do are Calamity Coyote, Little Beeper, Fowlmouth, and (seemingly) Gogo Dodo.
Babs gripes at this when wearing heels in "thirteensomething".