Sesame Street's Cookie Monster has been known to compulsively eat some rather odd things if cookies aren't available, including half a Volkswagen in The Movie, Follow That Bird. The Muppet Wiki has a list of things Cookie Monster has eaten.
Muppet monsters in general are Extreme Omnivores, but special mention goes to The Lunch Counter Monster and Carl the Big Mean Monster, whose sketches seem to revolve around the abnormal things they eat. Then there's Animal, who, amoung other things, once ate a TV set without unplugging it.
Joey on Friends, known for his voracious appetite, is effectively a walking garbage disposal. In one episode he catches Chandler and Rachel eating a cheesecake that fell on the hallway - and, without missing a beat, takes a spoon out of his coat pocket and joins in. Another episode has Rachel accidentally putting meat in a traditional English trifle. The others pretend to like it so as not to hurt her feelings, but Joey actually does - so much so that he eats not only his but that which the others had not eaten!
"What's there not to like? Custard? Good. Jello? Good. Meat? Good!"
It's also been suggested in one episode that Joey eats books. Possibly they had foodstains on them.
Also, when Rachel spilled spaghetti on the floor in "Casa del Joey", she apologized and he says "What the hey!" and throws some on the floor as well, saying she shouldn't worry. She then throws some more on the floor and his response?
"See now you're just wasting it. It's still food."
And proceeds to ladle all of the dropped spaghetti back on his plate.
When he, Chandler, Monica and Phoebe were trapped in Monica's bedroom during Ross and Rachel's big fight after the copygirl incident, they got so hungry that they began eating the wax the girls were using on their legs. Joey didn't think it was that bad while the other three found it nauseating.
Fear Factor, natch. (Though it's debatable if it counts as this trope, if you're only eating live maggots on a dare/as a stunt/for money.)
Red Dwarf's Dave Lister will eat just about anything hot and fried, oblivious to standards of cleanliness and sanity. A triple fried egg sandwich with chili sauce and chutney featured prominently in one episode.
Another episode saw him happily eating a space weevil. Although to be fair he thought it was a crunchy king prawn...
This is a man for whom a typical breakfast consists of cornflakes with tabasco sauce and a sprinkling of grated raw onion, served with a glass of chilled vindaloo sauce.
He's also pretty indiscriminate in regards drinks, too. In the first episode of season 8 he mentions polishing off just about everything in the drinks cabinet, including advocaat, that "smeg-awful pink stuff" (windowlene- a kind of glass cleaner) and "that chartreuse green liquory thing" (swarfega- heavy-duty hand cleaner normally used to remove oil and grease).
However, (most of the time), he absolutely refuses to eat Pot Noodle; to the extent that when it once came down to Pot Noodle or a can of dog food, he chose the dog food.
James May of Top Gear nonchalantly took a drink of Jeremy Clarkson's "Manly V8 Smoothie", which was composed of raw beef (with bones), bovril, tabasco sauce, chilis, and "for added bite", a brick, blended in a blender powered by a Corvette's V8 engine. His nonchalance abruptly ended when the full effect of the taste hit. Hilarity Ensued. When May could speak and/or see again, he dubbed the concoction the "Bloody Awful".
A partial list of other things that James May has eaten or drunk: garlic wine, grape juice he squeezed with his own bare feet, wine he fermented in the boot of a Jag, a deep fried Mars bar, the contents of a spittoon in a wine bar which someone had stubbed out a cigarette in, a sample of a puddle of unknown fluid leaking out of a caravan (which turned out to be his own homebrew beer), a prawn sandwich on an aeroplanewhich he dropped on the floor, snake whiskey, a bull penis, and fermented shark. Of those, the only things that he found objectionable were the garlic wine and the deep-fried Mars bar (which had been fried in fish-and-chip oil). He did claim that it was the prawn sandwich that made him "catastrophically ill" on the day of that interview, and not the weekend he'd spent in Dublin with his Top Gear mates. For extra hilarity, that fermented shark? Made Gordon Ramsay throw up.
Jeremy Clarkson also apparently has a philosophy towards novel food items encapsulated thus: "I had a puffin last week, that's not delicious, but the point of eating it was because I'd never had one before. I had some whale, and [the man serving it] said, 'Would you like me to grate some puffin on that?' How do you say no?! Yes, go for it, grate some puffin on that."
Direct quote from an interview: "I would put anything in my mouth. [beat] Exc-" [audience begins laughing]
What about Bizarre Foods With Andrew Zimmern? Hosted by the Travel Channel, the titular host goes places to sample the various local cuisines. Some of the more spectacular episodes include: insects galore, chicken testicles, rooster combs, not-quite eggs (an egg that was collected from the inside of the chicken before it grew its shell... as if eggs in general aren't squicky enough...), fermented pork, a hallucinogenic drink made by tribes in Africa where the women chew on the leaves of a plant for a while then spit the masticated glop into a big pitcher, and, the thing that actually beat the host, Stinky Tofu (tofu that is left basically to rot for days, gets fried up to make bread, and mushed together to make the filling). The host couldn't even choke back a single bite. He actually went and hugged the old lady who had cooked for him, and said, "You're the first chef to beat me." He claimed it tasted of raw sewage.
He also couldn't eat the durian.
He can eat it. He just thinks it's absolutely atrocious and refuses to eat any more of it because of the smell. It's not surprising, considering just how pungent durian fruits are.
No Reservations and its predecessor A Cook's Tour. Fellow chef and TV host Anthony "Tony" Bourdain once ate a cobra's beating heart among other things (see show entry for more details). That said, while he's friends with Zimmern, he criticizes his style in which Zimmern presents the food for the shock factor above its other qualities; Bourdain tries to portray the food with respect to its culture and origins.
All of the Jackass team are like this to some extent, although Steve-O and Dave England are perhaps the biggest examples - in particular, the latter ate horse manure in a $200 bet with the Three Six Mafia in Jackass: Number Two.
In the It's Always Sunny in Philadelphia episode "Who Pooped The Bed?", a forensic analyst dissects one of the poops in their quest to find out the episode's Driving Question. He determines that whoever did it was eating newspaper, as well as pieces of a credit card. It turns out that that still doesn't narrow it down from Frank and Charlie.
Charlie has a tendency to just eat inedible things and not be affected by them at all. Such items include an entire pear (including the sticker) and sunscreen.
Vyvyan in The Young Ones is shown eating, among other things, teabags and bricks. In "The Television Inspector", he literally eats the television to hide it from the television inspector:
Television Inspector: Aha! So you do have one, you little runt! The old trick, eh? Eat the telly before I get a chance to nic ya!
Vyvyan (electric cord hanging from mouth): It's a toaster.
Television Inspector: It's a telly, you yobbo!
Part of the mission of the chefs of Future Food is encouraging people to expand their definitions of what is edible.
In the Monty Python's Flying Circus sketch "The First Man to Jump the Channel", the titular character, played by Terry Jones, also intends to be "the first man ever to eat an entire Anglican cathedral". (He fails.)
iCarly: Sam Puckett eats many different forms and amounts of food including chugging of pickle juice.
M*A*S*H: In one episode, Klinger's attempt of the week to get a Section 8 discharge is to declare that he's going to eat a jeep. He is seen on screen consuming the windshield wipers and swallowing a bolt dipped in motor oil. (Reality Ensues quickly - he gets very sick.)
More than one death in 1000 Ways to Die involves someone eating something they shouldn't and dying as a result. I.e, there's the French nobleman in Bank Ruptured who has pica and eats metal regularly, which cuts the veins and arteries in his stomach and ends up killing him.
One Body of the Week on CSI had a disorder where his brain wouldn't process full signals from his stomach, leading him to be constantly hungry and trying to eat all sorts of things. He is shown noshing on pictures of food because he has already eaten his scheduled meal.
Jeff on Chuck had no compunctions against taking a bite out of a urinal cake in order to win a bet. The worst part is he didn't even need to, just be the first to touch it.
The Leviathans from Supernatural are heavily implied to be literally omnivotes, as Death claims God locked them away in Purgatory to prevent them from swallowing "the entire petri dish" (i. e., the entire universe).
On ''Cutthroat Kitchen, the Bobs appear to be this. In one Camp Cutthroat episode, a chef was sabotaged to have their basket swapped for the Bobs' bagged lunches. While some lunches contained food, a few others contained things like rocks or bundles of sticks. Yummy.