Bart often mockingly calls his father (Homer) by his given name, usually when he's bumbling something or just to mock him. It's a Running Gag in "Lisa's First Word" that this is a trend that goes back to him being a baby. (Lisa does it too at the end of the episode, though she calls him "Dad" in the modern day.)
Homer: Can you say "dad-dy?"
Homer: No, no, "dad-dy."
Homer: Why, you little—
When people are interviewed by news anchorman Kent Brockman, they like to put a "Kent" somewhere in their answer. Chief Wiggum is often guilty of this, Kent.
Hank Scorpio has his employees refer to him by his first name, as a way of enforcing his Just the First Citizen status.
At the end of the movie So the Drama, Dr. Drakken, the Big Bad, finally remembers the name of Kim's sidekick; throughout the show, he always referred to Ron as something along the lines of "the buffoon whose name escapes me."
Ron jokingly mentions in "Go Team Go" that he's on a first name basis with his (or really, Kim's) enemy Shego, and that he doesn't even know her surname. The joke is muddled by implications that "Shego" is an alias and Shego's surname is hiding in plain sightnote A few episodes, including "Stop Team Go", imply her last name is actually "Go".
Most of the operatives on Codename: Kids Next Door refer to each other by number. If two refer to each other by first name, it's safe to assume they're particularly close (Numbers 3 and 4 call each other "Wally" and "Kuki" all the time) or the moment is particularly deep (Number 1 calls a discouraged Number 362 "Rachel" when she's ready to give up her job as Supreme Leader).
The Delightful Children from Down the Lane also address the operatives by their first names.
In the early episodes of Gargoyles, Goliath always addresses Elisa Maza as "Detective", though he does refer to her by name when she's not around. Later in the show, as their relationship develops into a more romantic one, he becomes more and more comfortable calling her by her first name. Of course, none of the other gargoyles have this problem; the younger ones call Elisa by name from the get-go, and Hudson usually refers to her as "lass". Though it's a bit different here, as it's shown early on that gargoyles traditionally don't have naming conventions, or names at all, like the way the humans do (Goliath and Demona were originally the only two with names, and the others got them early on, and only out of convenience for Elisa rather than any attachment to a particular name).
All throughout the first and second seasons of Avatar: The Last Airbender, Zuko only calls Aang "The Avatar." However, he begins to call him 'Aang' after his change of heart in season three. The first Gaang member he calls by name is Sokka (whom he had previously called "you Water Tribe peasant"), followed closely by Appa, both in "The Boiling Rock." Once Zuko becomes Firelord, only his friends, his family, and Mai continue to address him by his first name. Even Korra calls him Lord Zuko and she calls the rest of the adults in her life (like Tenzin and Katara) by their first names.
In Code Lyoko, the Prequel episode reveals that Jeremie and Ulrich went through this during their first adventure. Prior to X.A.N.A. getting involved, they barely knew each other and addressed each other by surname.
Almost all of the main cast calls Bonnibel Bubblegum by her last name. With the exception of Marceline who constantly addresses Bonnibel by her first namenote Finn and Jake does address PB as Bonnie once but only during her brief time of exile. This is one of the many hints of their previous relationship.
A few characters are subject to this, such as Rattles and Slink the Tough Costumers. The most egregious example is probably Emily, who is D.W.'s best friend and has appeared numerous times on the show, yet her surname is never revealed. Lampshaded in an Imagine Spot in "D.W. Flips" and by Buster in the episode "D.W. Swims with the Fishes".
Buster: And the winner is... Emily, whose last name I don't know!
Nadine was this until mid-Season 16, where her last name is actually revealed to be Flumberghast. Justified in that, well, she's an imaginary friend conjured up by a 4-year-old.
In Rick and Morty, Morty always calls Rick (his grandfather) by his first name. In contrast, his sister calls him "Grandpa" and his mother calls him "Dad".
Some characters in My Little Pony: Friendship Is Magic who seem to be One Name Only are actually this. For example, Trixie's name is "Trixie Lulamoon" but she's always called "Trixie" in-series. This is made noticeable by how most characters are referred to by both their first name and their full name.
The title character of the Oh Yeah! Cartoons short "Planet Kate" has a last name, Moon, but it is never mentioned aloud within the short and is only shown on the back of the map she drew for school that gets taken by the alien dogs and during the short's end credits.
Steven Universe: Contrasting with the formality all other Homeworld Gems have toward them, the Diamonds just refer to each other by color. In "A Single Pale Rose", this is another way that the viewers are tipped off that "Rose Quartz" was in fact Pink Diamond, when she refers to the Diamonds with simply "Blue" and "Yellow".
The Doom Patrol in Teen Titans are the only superhero characters in the show that use their real names. They're like a family after all. At the end of "Homecoming Part 2", Elastigirl (whose name is "Rita") calls Beast Boy by his real name "Garfield". Not even Beast Boy's teammates knew his actual name (and probably for more reasons than one).
On The Fairly Oddparents, an episode revealed that Crocker was friends with his current boss, Principal Waxaplax, in their childhood. During this time Waxaplax was in love with him. This ended with his claims of fairies in college, which got worse over the years. By the time he was starting his teaching job, Waxaplax was demanding to be called by her last name.
Regardless of how romantic they get, Fry and Leela never refer to each other by their first names (Philip and Turanga, respectively). It's tricky on Leela's part, since Leela's "first" name is her family name — her parents are Turanga Morris and Turanga Munda, but it's unknown if Leela was aware of the naming order since she was "abandoned" at a ophanarium and didn't meet her parents until some two decades later, and the viewers weren't clued in on it until they were introduced, so for a while it was thought Turanga was her first name. Regardless, he's still referring to her by her given name, so that's not as strong an example. Fry, however, is. Up until "The Problem With Popplers", Fry didn't know Leela's family name, and Bender didn't know Fry's given name.
There was also an episode with a past girlfriend coming to the future who only referred to Fry by his last name Fry. It seems to be that the only people who called him Phillip or Phil were his parents and brother.
Professor Farnsworth is usually called "Professor" by Fry, who is his great great great etc. uncle. Pretty much the only one who gets to call the Professor "Hubert" is Mom. Rarely, Zoidberg also calls him "Hubert" after it was established that the two have been friends for decades (and on very rare occasions, the Professor has called Zoidberg "John").
There was an episode where a Straw Feminist comes to the school after Skinner made a sexist remark. While talking to her, Chalmers uses her first name and she replies that it's a form of harassment, so he decides to go all out and kiss her.
A completely non-romantic example is present in "Two Bad Neighbors", between Bart and former President George H.W. Bush.
Bush: You know, in my day, little boys didn't call their elders by their first name. Bart: Yeah? Well, welcome to the 20th century, George. Bush: I'll kick you right out of the 20th century, you little...
In "Who Shot Mr. Burns?" Burns and Smithers have an exchange where they call each other "Waylon" and "Monty." Since they always refer to each other by last names only, this was probably to emphasize how personal it was when Burns fired Smithers shortly thereafter.
In "Whacking Day," when Marge attempts to home-school an incorrigible Bart, she introduces herself as Mrs. Simpson, then stifles a chuckle.
Subverted with Professor Utonium and Ms. Keane on The Powerpuff Girls, as neither character's first name is ever disclosed. The Girls themselves are mentioned by first name only but never suffixed with Utonium, though dialogue does mention that "Utonium" is their surname.