Gag Lips

So that's what he looks like beneath his mask. Huh.

Until recently, large lips were not considered an attractive feature (and even nowadays Your Mileage May Vary on when plushy lips stop being attractive and become grotesque), so naturally, big lips are lumped with other physical features considered "ugly" by the fashion zeitgeist.

Extremely large lips have a long story of being used in a humorous manner. The most common methods in media are:

To create in instant Gonk.
To mock Western civilization's obsession with plump pouters.
To have a visual cue of the after effects of any Blazing Inferno Hellfire Sauce to come along with the Fire Breathing
To serve as a symbol of hypersexualization (the "porn actress lips").
To portray people of African descent. Note that this use is strongly not recommended nowadays.

Compare with Gag Boobs and Gag Penis for the trifecta of oversized body parts used for humor.

Examples of Gag Lips include:

Anime and Manga

Naruto and company suspect that Kakashi might have this; their Imagine Spot gives us the page image.

Art

Salvador Dali's Mae West Lips Sofa.

Comic Books

Hildagaard Rail in the story "Sky Highway" from the original Teenage Mutant Ninja Turtles comic book series, #30. She was a mutated German woman who had lips so big they took up most of the space on her face, going practically from ear to ear and forehead to chin. Otherwise she was very well endowed, had long legs and an hourglass figure. When Raphael attempted to leave Hilda she grabbed him by the tails of his bandana and gave him a giant kiss goodbye against his wishes. The lipstick mark she left on his face covered his entire beak.

Film

Gremlins 2: The female gremlin has ridiculously large lips.
In The First Wives Club, Goldie Hawn's character gets an overdone collagen injection.
In Hercules, Pain and Panic are sent to stop Icarus from kissing Cassandra (because that would violate her deal with Hades). One thing they try is to give him ridiculously large lips in an attempt to embarrass him out of kissing her. Instead, it just primes him up all the more.

Live-Action TV

Just Shoot Me: Nina is invited to a talk show to talk about lipstick, but before she goes on, her peanut allergies act up after she eats some candy at the green room, giving her puffed-up lips and Elmuh Fudd Syndwome.
There is a Russian series called "Odna za Vseh", meaning roughly "one woman who does everyone's job". Each episode consists of a string of short sketches, where the same actress plays a dozen vastly different roles. One of these is that of a millionaire's wife. Both she and her best friend have lips that look like they should be able to deflate them for storage.
On Nick Arcade, landing on an enemy space in the Cape Cosmos board would result in Mikey being approached by the Smooch Aliens, a pair of round aliens with pigtails, cyclopic eyes, and big, red lips, who would give him slurpy kisses.

Music

The Rolling Stones' logo, complete with gag tongue.

Newspaper Comics

In one Zits strip, Jeremy's lips swell up after he is stung by a bee while riding his bike. Cue the girls cursing how lucky he is and wondering whether they should take up beekeeping.

Video Games

Jynx and Smoochum from Pokémon qualify. Some of their powers are centered on their lips, such as their attacks that involve kissing, and the Pokédex describes Smoochum as exploring the world using their sensitive lips. This is one of the reasons Jynx's appearance invokes Unfortunate Implications because of her resemblance to blackface or "mammy" characters.
- ... Even when the Jynxs seem to be actually inspired by the Ganguro/Yamanba subculture, where Japanese girls show off their rebelliousness against society via bleaching their hair and adquiring deep tans. Said trend/subculture was very popular in the early Pokemon years.

Web Comics

A constant mockery/Self-Deprecation point of Andrew Hussie, who has large, pursed lips. While in Homestuck he's drawn without a mouth or nose, his title is still "Huss of Lips", and references are made to his Author Avatar, Dave, having hilariously large lips.

Web Original

Gaia Online has a ghost of a supermodel equipped with the wax lips items, suggesting she's hit the collagen a bit too hard.

Western Animation

Back when Africans were Acceptable Targets, this was a fairly common sight in children's cartoons.
Lena Hyena from her own cartoons and Who Framed Roger Rabbit??
On SpongeBob SquarePants, Sandy has to battle her way through Karate Island in order to save SpongeBob. One of the minibosses she has to fight uses her lips as weapons. Sandy defeats her by drying her lips with a hair dryer until they are cracked.
Gorgeous Gal in the Woody Woodpecker cartoon A Fine Feathered Frenzy. Since she is a bird her lips only appear big at the end of her beak whenever she puckers them or attempts to kiss Woody Woodpecker. At one point Gorgeous Gal appears in a bath and playfully blows bubbles at Woody and her lips are about the size of his face. Another time she rushes to kiss him with the full force of her lips and misses, kissing the wall. The lipstick mark she leaves on the wall is almost the size of Woody's entire body. Whenever Gorgeous Gal does manage to kiss Woody her lips are shown to be much bigger than the Woodpecker's as well.
In the Captain N: The Game Master episode entitled I Wish I Was a Wombatman a beautiful director's assistant who looked like a short pink octopus version of Dolly Parton had the hots for Simon Belmont. When she hugged him with four tentacles around his neck and arms she gave him a wink and extended her giant red lips in front of his for smooching. Simon didn't reciprocate. Eventually the busomy babe chased after the object of her heart's desire offering to do his make-up. Simon ran because it was pretty obvious that the octopus really wanted to wrap him in her tentacles and shower him with kisses.
Johnny Test: Johnny test out his sister's Lip0 Lozenge, designed to give fuller, kissable lips. It works too well.
This was the result of an attempt to stop a Zombie Apocalypse in a recent^[when?] episode of Adventure Time.

Multimedia

This occurs more often in a depiction of a Gonk.

Real Life

The infamous "Duckface" is an unintentional example. It is done by girls and sometimes boys by puckering their lips to make them look more attractive, but the result looks something like a duck's bill. It's become almost a Running Gag on sites like Facebook.