The Tasmanian Devil in Looney Tunes is the animated archetype for this character. Many of the jokes in the WB shorts featuring him involve other characters easily tricking him into eating something that no sane creature would, like burning sticks of dynamite, inflatable life rafts, and in one case, an entire refridgerator.
Likewise with the spin off series Tiny Toon Adventures with the character of Dizzy Devil, a younger Tasmanian Devil. One episode even featured the characters having a contest, "Will Dizzy Eat it?" where Dizzy was fed various things including a jet plane.
Taz's entire family has shown signs of this trope, although none of them to the extent that Taz is.
Jim from Mission Hill was constantly eating everything in sight, including leftover food in restaurants and a chunk of the apartment sofa.
Stoagie falls under this as well. Then again, he's a dog. So it's probably within the realm of normal. In fact, Jim eats a few things he saw Stoagie eating first, reasoning that if he liked it, it can't be that bad.
The Archmage on Gargoyles ate the Grimotum Achenorum in order to subvert a rule that Avalon has about bringing human magic onto the island.
Cyborg, in the Teen Titans episode Crash is infected with a computer virus, making him perceive objects and various people as some of his favorite foods. Normally, Cyborg is a Big Eater. Mayhem and Hilarity Ensues.
Starfire exhibits such a behavior occasionally, though usually out of ignorance of what Earthlings consider food (or out of the habits borne of some of the really gross things that Tamaraneans do consider food). For instance, she considers mustard to be a most delicious beverage.
Also, in the episode he was introduced, Silkie ate his way through half the contents of Titans Tower.
Terra, too, shows herself to be one. For one thing, she's the only person to stomach Starfire's food and like it!
Gonard from Kappa Mikey alternates between a Big Eater and an Extreme Omnivore on a whim, and is generally the epitome of either (when not the epitome of both). During one scene, he took video cameras from people, put them on a sandwich, and, at the end of the chase scene, ate it on one bite.
He basically will eat anything if it's put into a sandwich.
South Park has Kenny, who is not against consuming anything if dared, offered money for, or to get high. Amongst his exploits is drinking gasoline and getting high on cat urine.
Ed from Ed, Edd n Eddy apparently never listened to the old adage "never eat anything bigger than your head": when sitting down to a bowl of soup, he eats the bowl as well, and Eddy once tricked Ed into eating his bed.
Ed: [wipes his mouth after eating the bed] That hit the spot!
A Running Gag in Drawn Together involving Toot is her habit of eating literally anything that she can grasp (she is a one-dimensional fat joke, after all). She has eaten remote controls, refrigerators, large automobiles and even people (she has eaten at least two housemates on separate occasions).
Big Dog in 2 Stupid Dogs is pretty much capable of eating anything he would want to eat, in one occasion even functioning as a canine vacuum cleaner. His eating habits are comparable to Guu (mentioned above), in a way that anything he eats is capable of coming out again in one piece (if mostly covered in saliva). Unless he actually chewed, of course.
Little Dog: I just saw an airplane!
Big Dog: I can eat an airplane.
The cartoon Fat Dog Mendoza stars the titular dog (who's actually more spherical than fat) and literally stores things inside himself.
One episode of The Life and Times of Juniper Lee featured a creature known as a "Batoot", which ate anything in its path, and had the added misery of bringing bad luck with it wherever it went, the severity of which was measured by its unhappiness. The more miserable it became, the worse the circumstances around it became. When Juniper was supposed to take care of it until its herd returned, the large green (but somehow endearing-looking) creature devoured the entire contents of both the attic and the basement of Juniper's home.
Homer Simpson from The Simpsons has been known to eat some very weird stuff. He's eaten stuff like bacon mixed with fudge, butter in his coffee, a horse head, and a bag of flour, He's also eaten some inedible stuff like a pair of mens' slacks (which apparently tasted better than Indian cuisine), a bowl of change, a jar of petroleum jelly, radioactive material, and a rubber biter covered in petroleum jelly. However, he doesn't like eating anything that's low in fat.
In one episode, he ate the baking soda that had been in the back of the fridge before they even first moved in (before subsequently going on an 'antacid trip'). Lisa is then revealed to be on a first name basis with the Poison Control Hotline operator.
Bart tries to do this to win a contest where you try to humiliate yourself as much as possible, but the audience proves too much for him ("Please, no more spark plugs.")
Ralph Wiggum occasionally eats glue and crayons. He even made a "crayon sandwich" for a cooking contest.
Dr. Zoidberg of Futurama has been known to eat anything he can get his claws on, including a boot from the ocean floor, a weeks-old moldy sandwich, a tiny edible crown (made of wood), cholesterol scraped from one of Fry's arteries, the Earth Flag, a deviled egg surgically extracted from his stomach and a live raccoon he found in his shell after fishing it out of a dumpster. He has also expressed curiosity as to what the Shroud of Turin tastes like.
His race was single-handedly responsible for the extinction of the anchovy.
Nibbler. Oh, God, Nibbler.
Nibbler and the Niblonians are a better example of Big Eater while they eat huge quantities, unlike an Extreme Omnivore they generally spit up things that would be considered inedible such as animal collars, non-edible parts of an animal (such as antlers) etc. Zoidberg by contrast eats shoes, garbage, and other things clearly not meant to be food.
All of the Niblonians for that matter. Feast of a Thousand Beasts anyone?
The titular character of Chowder can be classified as this. He's known to eat other things than food such as maps and even once eating a marching band (but he let them go soon afterwards)
Oddly, Anakin in the Star Wars: Clone Wars miniseries brings a bag of bugs and worms to the little makeshift shelter he and Obi-Wan hang out in while waiting out a protracted siege and starts happily chomping on them. When Obi-Wan expresses his disgust, Anakin quips about how Obi-Wan always taught him to "feed off the living Force".
Beezy and his girlfriend Saffi on Jimmy Two-Shoes are known to eat just about anything, sometimes snatching it up frog-like with their tounges.
Stitch's appetite expanded from just eating alien food in the movie, to gobbling anything that strikes his fancy in Lilo & Stitch: The Series. Throughout the series he's eaten garbage, tools, nuts and bolts, a can of macadamia nuts, a left shoe, several cans of soda, Mertle's awful rock-hard muffins, a container of fresh eggs and even Mertle's dog (thought he spit it out later).
There was also moments of extreme eating from Stitch's cousins. In one episode there was an experiment named Tank that was solely designed to eat metal and grow bigger and an experiment named Wishy-Washy who was briefly seen eating a pack of comic books
One Goofy episode from Disney's House of Mouse had Goofy learning etiquette. At the dining table he eats the meal set in front of him, and then eats his metal utensils, plate, and table (he justifies that he needed the iron).
Owen from Total Drama, whose steel-clad stomach (downing everything from Chef Hatchet's food, cockroaches, bull testicles, and even dog food) landed him in the finals. And he won!
Yet he got pretty ill in Total Drama Action after eating an entire buffet of foam-core food (although he thought it was real food and had it all devoured before anyone could tell him otherwise). He still nevertheless found it delicious.
Sugar in Total Drama Pahkitew Island, being not only an Expy of Owen but Honey Boo Boo as well, has taken up the mantle. Her stomach is even stronger than Owen's!
In Justice League, one of John Stewart's fellow Green Lanterns, Kilowog, is shown to apparently be this in a scene where he and Flash were poking around John's apartment. Flash first finds a carton of "Bob and Terry's" ice cream in the freezer and shows it to Kilowog, who tosses the entire carton into his mouth, eats it, and declares it "delicious!" Flash then continues searching and finds a VHS tape of Old Yeller. He gives it to Kilowog just to show him, but once again, Kilowog tosses it in his mouth and eats it, declaring it "delicious!"
One episode of Bonkers dealt with Bonkers and Lucky re-institutionalizing a hairy beast known as The Louse who devoured everything in sight, when Lucky tries to better his table manners he eats the entire table along with the plate and eating utensils.
Pinkie Pie has shown a willingness to eat virtually anything, eating cupcakes with hot sauce and ashen lumps that are supposedly cupcakes without complaint. Her first reaction to seeing a vat of liquid rainbow was to taste it, and it's the only thing she's complained about the flavor of on-screen — although she had some bizarre muffins that made her sick, she didn't actually comment on their flavor. So she's probably an Extreme Herbivore at best.
Her sister Maud has no problem crunching into hunks of crystal, and she taught Pinkie the family's rock candy recipe - which uses actual rocks as one ingredient.
Spike from the same definitely qualifies — he prefers eating gemstones, but can and often will eat literally anything. Including eating and enjoying said muffins. After seeing a live earthworm stick its head out of one.
The alien race in Ben 10 referred to as 'Gourmands' (or, as Ben himself calls said form, 'Upchuck') are essentially an entire species that are this trope, with the added benefit of weaponising what they eat by spitting it back at their enemies with great force. Amusingly, they seem to be incapable processing anything which would be edible to a human.
This is also the basis of their plan when facing planetary conquest. After all, it's very hard to take over a world when the inhabitants are willing to eat the planet and use it as fuel to find a new one. Their planet is (or rather, was) called Peptos XI for a reason.
Chris from Dan Vs. It takes him a while to register that he's eaten something human beings really shouldn't eat.
Ringo Starr in The Beatles. In one episode, George Harrison complains that he's so hungry he could eat his guitar, and Ringo tells him in all seriousness that guitars taste terrible and offers him a (literal) drumstick instead. George gets angry and shoves the drumstick into his mouth, and he shrugs his shoulders and eats it.
Gravity Falls: Implied in "Fight Fighters" when we see one of Mabel's drinks contains a strange cyclops troll doll and several dice, all in a green fluid.
Mabel also once ate an entire tube of toothpaste, just because it was sparkly, and according to a Reddit discussion with series creator Alex Hirsch, she was sent to the hospital after attempting to eat scratch-and-sniff stickers.
The Insecticons of The Transformers are capable consuming both organic and inorganic matter. A field of grain, a metal blast door, or even an experimental power generator are all fair game.
In He-Man and the Masters of the Universe (1983), Skeletor's henchman Trap-Jaw could eat anything, using his metal jaw to chew up rocks and metal. He was also able to gain Super Strength in one episode of the original series by eating an enchanted mineral called Eternium. (He believed he could do so in the remake as well, but he never succeeded in obtaining any; however, when tricked into eating something called "deuterium", he became sick.)
Entree of Spliced once challenged someone to see who could eat more toolboxes. Another time, he ate a rhino. Not rhino meat-a whole, living rhinoceros. In fact, he's eaten most of the other characters at some point.