Subverted, handwaved and lampshaded a lot in various Muppet productions, usually involving the Swedish Chef:
In an episode of The Muppet Show the Chef tried to make a Thanksgiving dinner. Trying to make turkey doesn't work since the turkeys can talk; trying to make pig stew is as bad, with pigs in the cast. He goes for "veggy weggy stew" but the vegetables can talk and fight him. In the concluding scene the Thanksgiving dinner consists entirely of vitamin pills.
In another episode he tried to boil a Lobster until the Lobster's brothers rode in Mexican-Bandito style, shooting up the kitchen with their revolvers and rescuing the main course.
He had a similar problem making Christmas dinner for A Muppet Family Christmas. The Chef invited the Turkey (from Dorchester, MA) for dinner. The Turkey convinces the Chef to roast up Big Bird instead. Big Bird unwittingly saves his own life by befriending the Chef, and in the words of this reviewer, "The Chef ends up preparing shredded wheat and cranberry sauce, which is terrific. Until the cranberries start singing 'Silent Night'..."
A good way hit Miss Piggy's Berserk Button is to mention any kind of edible pork products in her hearing.
In The Muppet Christmas Carol, Rizzo the Rat is about to eat some vegetables prepared by the Swedish Chef...but they join in song, and he shakes his head and relates his mother's advice: "Never eat singing food."
Then there was the frog's legs skit, the duck soup episode...once his spaghetti tried to crawl away from the plate while he was checking on the tomato sauce, and ended up attacking him when he slapped it back. Another time, bread dough started inflating and finally took him over. And each and every time it was absolutely hilarious.
In Muppet Treasure Island, Mrs Bubberidge the innkeeper announces tomorrow's special is "roast suckling..." and when the appalled pigs turn to her, she concludes "...potatoes". And then she has to apologise to a talking potato...
In the Julie Andrews episode number The Lonely Goatherd, one must wonder what the lonely goat herded (especially since no flock was shown on screen).
Well, no wonder he was lonely!
Averted altogether on Sesame Street, where the issue is simply avoided. When Elmo inquired about what wild animals eat, he was told about herbivores and insectivores, but meat-eaters were never discussed, even though (talking) tiger and lion Muppets were right next to him.
There was a Tales from the Darkside episode called "Your Weight Is Over" that took this concept to the very extreme. A malevolent "diet company" gave a woman the power to hear food talking. Any food, vegetable, or animal. So whenever she bit in, it screamed. She starved to death in the end.
Puzzling, in as much as fruits (and many so-called vegetables, such as tomatoes and eggplant) are not whole organisms; they are in effect fertilized ovaries, deliberately cast off by a plant in order to facilitate its reproduction. Even if you pluck an apple from a tree instead of waiting for it to fall, you're not killing (or indeed hurting) anything. The seeds are designed to pass through an animal digestive tract unharmed and viable.
If you think about that too hard, especially in the context of sentient apple trees, you get a whole different kind of Squick (see Equal Rites).
Processed meat certainly wouldn't be able to speak, either, so both must just be a trick.
Fresh vegetables can feel "pain" much better than processed meat. Animals feel pain via nerve cells, which are dead at that point. The vegetable's equivalent damage-recognition system (which is based on hormone changes and cell membrane voltage) is fully functioning while it's fresh. Vegetables do not have central nervous systems to process the sensation, so the point is moot unless you have an aversion to damaging food in general.
An especially odd example: During one of his Headlines segments, Jay Leno showed a newspaper article with tips for a fun camp-out. For maximum strangeness, the picture for the article showed three anthropomorphic marshmallows roasting an inanimate marshmallow over a fire.
Possibly the most disturbing (and, yes, Nightmare Fuel) variation of this trope: TV Funhouse. The whole show is a satire of kid's shows. Except for one human, most of the cast are animals. Some of the animals are played by puppets, others by actual animals. In one episode, all the animals went to this restaurant where the whole gimmick was that you eat what you are. The cats can eat cats, lobsters eat lobsters, and so on. Get ready for this, because I'm leading up to the disturbing part, which will be the very last three words of this entry. Remember, some of the animals were played by puppets, and others were played by real animals. And the theme of the restaurant was that you eat your own species. Well, they had a real life pig, and real food, and they showed the pig eating bacon.
Pigs are omnivorous and happily eat anything. It's not disturbing that they fed a pig bacon, as non-sapient creatures have no morals to have qualms with cannibalism, anymore than the rest of the scene is disturbing.
And among livestock, pigs are infamous for cannibalism, especially eating their own young. Many other animals do this as wellnote even animals that are normally herbivores, such as rabbits, particularly when under stress from overpopulation or malnutrition.
So you know that bit at the beginning of The Wizard of Oz where Dorothy falls into the pigpen and everyone panics? The reason for that is because it's fairly common for pigs to eat their keepers if given the opportunity. A pig eating bacon is barely a squick.
Also, in case you weren't aware of this, there are plenty of animal farms that feed suet (leftover bonemeal and unedible or unwanted animal parts) back to the animals. In some countries this is illegal or at least restricted (e.g. feed cow suet to the pigs, pig suet to the cows, or somesuch). The main reason to avoid this (like cannibalism, aside from the moral implications) is that same-species feeding makes it easier to contract diseases from the food; most notably Prion Disease, aka "Mad Cow Disease". The EU has adopted regulations prohibiting or strictly regulating certain types of animal products from animal feed for this reason.
One sketch on The Young Ones had a couple of rat puppets chatting in the background. Spotting them, and apparently not understanding their language, Rick smacks one with Neil's guitar, killing it. A quick scene of the surviving rat eating the dead one follows, in which it Hand Waves this trope, remarking: "It's what he would have wanted".
On Arrested Development the Bluths' Frozen Banana stand has as its mascot Mr. Banana Grabber, a giant talking banana that steals and eats regular-sized, non-talking bananas. The implications are not lost on Michael.
Michael: Why would a banana grab another banana? I mean, those are the kind of questions I don't want to answer.
In a rather silly example, Animal Planet's Weird, True and Freaky featured a story on chimpanzee "cannibalism." Anyone familiar with chimps knows that the males tend to kidnap and eat the infants of rival troops. Shocking, yes, but that's not what the story is about. It's about chimpanzees, apes, hunting for monkey meat. And the narrator says it's cannibalism. That's as cannibalistic as a polar bear eating a seal! It also reaches Unfortunate Implications, as many people in Africa regularly eat bushmeat (see the real life examples below)