By far the most common approach, especially in older fiction, is the Predators Are Mean subtrope. All your heroes are herbivores. All the villains are carnivores à la The Big Bad Wolf from the Three Little Pigs.note In Real Life, pigs and wolves are both omnivores, eating both plants and other animals. Wolves will prey upon pigs, but only to the extent that they would prey on other organisms, e.g. it would be rewarding and they could pull it off without getting themselves killed. This subtrope is so strong culturally that people assume Real Life predators are mean, evil, and nasty, and all the herbivores are cute, cuddly, and friendly. Anyone with any knowledge of real animal behavior knows that's not the case at all; some herbivores like rhinos and hippos can be extremely aggressive, while social predators (wolves, orcas, etc.) can be tamed, trained, and even become companions.
Sometimes your heroes are predatory animals. Now carnivores are okay, so scavengers often become the Villains By Default. The usual ethos is that only evil weirdos eat carrion, and only cowards do not hunt. They will be depicted as ugly and intimidating. Often the carnivorous heroes will never been seen actually killing and/or eating another animal. Example: The Lion King.note Non-exclusive scavenging and predation is rarely used in fiction. In Real Life, lions scavenge off hyena kill more often than vice versa.
A increasingly popular option in fiction has been to render the carnivore's prey in a realistic, non-cute manner. The prey does not talk — the prey is not humanised in any way. Fish, in particular, are nearly always a viable mealtime option, unless they're major characters.
Similar to the option immediately above, some works such as The Chronicles of Narnia and the Spellsinger novels make it clear that only some of the animals have human-like intelligence. In some fiction, there are explicit differences between the anthropomorphic and normal members of the same species — bipedality, speech, clothing, etc. It's okay for a talking lion to eat a non-talking deer, but eating a talking deer would be tantamount to cannibalism.
Establish that the carnivore is unlucky (Wile E. Coyote) or that their chosen prey is too fast or aggressive to catch — for example, Jerry of Tom and Jerry. This makes the point moot, since we never see the predator eat.
Invert most of the above examples by making the prey totally unlikable. The cat gets double satisfaction when eating that jerkass mouse, as not only is he delicious, it serves the little bastard right.
Somewhat similarly, the solution in works with more of a science fiction bent is that the technology available has created meat substitutes that are readily available for humans and animal carnivores. (Star Trek: The Next Generation explicitly pointed this out in an early episode.)
Others will play it up for Refuge in Audacity. The pig knows enjoying a nice ham dinner is cannibalistic, but just doesn't care.
Still others, usually ones that take place in a World of Funny Animals, will completely ignore the issue altogether. The cow will happily join his dog friend for some burgers, and absolutely no-one will think of the implications.
Finally, predation can be treated as just a fact of life. See Rudyard Kipling's The Jungle Book, the Dinotopia books, and the WebcomicKevin & Kell, among a very, very few others: Carnivorism happens, it's nature, and it may even be incorporated as a part of both the talking animal economy and social structure.
So are the cereal squares in commercials for Golden Grahams.
Charlie the Tuna who always wants to be apart of Star-Kist Tuna. Much to his chagrin, he's never good enough to be killed, chopped up, and eaten. Sorry, Charlie.
A newer Lay's commercial has Mr. Potato Head coming home from work to catch his wife Mrs. Potato Head eating some chips.
Pagliacci features a restaurant run by griffins aimed specifically at carnivores. As can be expected, they serve meat items, including beef. However, this is set in the world of My Little Pony: Friendship Is Magic... where cows are presented as sentient creatures...
Taken to the extreme in Thorbjørn Egner's Dyrene i Hakkebakkeskogen (The Animals Of Hakkebakke Forest), one of Norway's most popular children's plays ever. By popular vote, the animals pass a law banning carnivorism. One of the animals who supports the law is the alpha bear (brown bears admittedly eating plants for 80% of their diet), while one of the animals who opposes it is a hedgehog (which eats invertebrates and small vertebrates).
Carnivorism is played much like alcoholism here too, as the fox, who had been the main antagonist up to this point, turns to stealing meat from the nearest farm to get his "fix".
In the Sylvanian Families franchise, there are a lot of food accessory sets, and not all of them are clearly vegetarian. One release is a fish and chips van, and some of the highly detailed food can be a little suspect. Barbequeues are also risky. But topping that is the hamburger resturaunt, and for those low on space there's also a hamburger wagon.
Official Babble: No fast food rubbish here! All the meals on offer at the Sylvanian Hamburger Restaurant are top quality, healthy, locally sourced products! If you're in a hurry, the restaurant even has a take-away window that you can drive up to!
Official Babble: The Hamburger Wagon has fold out counters and fold out rooftop seating area. Comes with lots of hamburgers, hotdogs, sandwiches, rolls, drinks and other accessories.
A Farmyard Accessories set comes with milk churns. There was a milkman figure (currently unavailable). And finally, there's a cow family. For the contemplative among us, this borders on Squick material.
It's worth noting that in the 1987 animated series, the michievous 'Slydale' fox family was the most likely to cause trouble, using Predators Are Mean. Even now rabbit families outnumber any other species.
Darwin's Soldiers mostly avoided this trope for the first RP as all meat was fish. The second RP got a little sticky with Aisha wanting a meatball sub and slabs of meat hanging in the walk in freezer. The third RP also mostly avoided it as well.
However, Word of God is inconsistent on this matter; Serris said that anthro and "normal" animals exist and eating "normal" animals was acceptable. But he has also said to assume all meat is seafood.
The Furtopian Hoofer Revolution embodies this trope. The entire hoofer revolution of 2010's April fool's day forum take-over revolved around this trope as its theme; the ungulates and other herbivores rebelling against the carnivores. Further hilarity ensued once a herbivorous fox, a carnivorous rabbit, and a number of shape-shifters came into play, adding to the confusion.
Inverted in ASDF Movie 7, where the talking muffin wants to be eaten (and is so cheerful about it!) yet nobody he meets feels like eating him.