Fables takes the "only some animals can talk" route. The Big Bad Wolf, sending his children out for some hunting practice, tells them to be sure to chase the animal long enough to make sure it isn't sapient before they kill it. Not all the carnivores will do this though. There are some that have no issues with eating talking animals and Bigby himself used to be like this, so this is more an issue of personal ethics. The problems seem to come when the prey are recognized citizens of Fabletown and so protected by its laws.
In the short-lived furry comic book, SpaceWolf (Not that one!), the characters are of intelligent furry species that still have the predator/prey relationship such a Wolf species raiding a planet populated by humanoid sheep. Eventually, the sheep rebel and a long war starts that is only resolved when the sheep develop soy-curd meat substitutes which satisfy the predator species' needs. However much later, a villainous Sheep monarch decides to seize power by first demonstrating the lethality of an awesome planet destroying weapon. As terrible as that is, that is actually the lesser of his two major threats to cow the interstellar population: he also threatens to cut off the flow of soya-cord food to any resisting planet and let it fall back into murderous predator/prey chaos.
That raises the question of what the wolves were eating before they invented spaceflight.
They ate sheep from their own planet. That's why they are extinct, and why they have to go to space now.
The DC Comics series Captain Carrot and His Amazing Zoo Crew! takes place on Earth-C, a parallel Earth populated solely by anthropomorphic animals. The series presents the world's populace as being vegetarian (though eggs are acceptable, and vegetarian versions of burgers, hot dogs, etc. exist), including carnivorous-in-real-life species such as felines. The comic explains that while their prehistoric, uncivilized/unevolved ancestors did eat meat, animal-kind ceased doing so once it became "civilized". A villain in one story attempts attacking the Zoo Crew with fierce prehistoric animals brought from, as he describes it, "the age of the flesh-eaters." The idea of eating another animal is viewed among Earth-C's populace as cannibalism, as seen in one story where a "wuz-wolf" — a friendly wolf who, under a full moon, becomes a feral human — attempts to eat a pig.
In Blacksad we see a photograph of a barbeque where two polar bears and an arctic fox (anthropomorphic) are grilling sausages. In this world not all animals are anthropomorphized...but the pigs are.
The French comic De cape et de crocs, which is set in a Lions and Tigers and Humans... Oh, My! world, plays with this trope. In one book, the main characters are taken prisoner by a tribe of savages who, at first, seem to be your average Cannibal Tribe: they bind them, dump them into a cauldron of boiling water and vegetables, the usual works. The heroes manage to free themselves and befriend the tribe leader, who is confused when they complain to him about the barbaric custom: his tribe, he says, have never been cannibals. That's when the characters seem to remember that they are, actually, an anthropomorphic fox and wolf. And the savages, while not cannibals, definitely have no problems with eating dog meat. (That's only one way in which the comic parodies and subverts various adventure tropes.)
Which doesn't mean that Carnivore Confusion isn't played straight elsewhere in the comic. The galley of the pirate ship definitely contains pork, when one member of the crew is a talking pig.
The newspaper comic Tom the Dancing Bug parodies this in a strip called "Anthropomorphic Antix". An anthropomorphic dog is walking a dog on a leash when an anthropomorphic pig holding a bag walks up. The pig says, "I don't get it. Aren't you both dogs?" to which the dog replies "Isn't that a pork chop in that bag?"
Douwe Dabbert features an animal kingdom populated by Funny Animals. Every single one of them is vegetarian, including normally predatory ones, such as the wolves. Two human villains then entered the kingdom. Not knowing that the animals were intelligent, they slaughtered and roasted the innkeeper, a chicken (the inn as empty at the time, and quite isolated from the rest of the kingdom). When the wolves arrive, they eat along, not knowing it's the innkeeper. When they realise it afterwards, they decide that they like the taste of meat, and proceed to conquer the kingdom with the human villains. They go back to being vegetarians at the end of the story.
ElfQuest doesn't really fit the "talking animal" topic, but it notably avoids the "carnivores are okay as long as they are predatory" bit. For the Wolfriders (a tribe of elves), it is normal to leave the bodies of their dead to the wolves.
In Snarfquest, after the gaggaleech/death leech Wished for the ability to communicate with any living thing, it discovered to its horror that a type of prey-animal it'd fed on previously was just a baby, and began screaming for its mother when the leech attacked. For the rest of the series, the gaggaleech wound up feeding on fruit (which still cried for help!) or the blood of fresh-killed monsters slain by Snarf and his friends.
In The Courageous Princess both talking and mundane animals can be found. Ones that can talk are given full human rights. Mundane ones can be eaten without any trouble. It happens that a talking animal is murdered via being silenced and tricking someone else into eating them.
The sentient animal society in Chlorophylle by Raymond Macherot has a lecteresque criminal mastermind who consumes other sentient animals
In Sam & Max: Freelance Police, while there's only a few truly anthropomorphic animals, lots of other animals are sapient and can talk. This might be deliberately dark humour to reinforce what a Crapsack World the setting is, though. In addition, Max is repeatedly shown eating meat, even though he's a rabbit, albeit one with carnivorous-looking teeth.
Played with in early issues of the children's magazine Cricket, where the marginal comic strip's talking insects hung around with Zoot the shrew: a mammal who, by rights, should've been eating them all. For a time, Zoot's friendship with "everybuggy" was justified by his being so nearsighted, he mistook them for bunnies and other creatures not on a shrew's menu; later, he got some glasses and learned the truth, but conveniently also turned vegetarian.
Beast Boy of the Teen Titans has come to this conclusion: as he can transform into any animal under the sun, eating any kind of meat, feels like cannibalism, with eggs having a squickyimplication. He tries to share his tofuburgers with the other Titans whenever possible (they rarely partake), and has been known to turn into particularly cute animals and quip that they should at least vow never to eat green meat.
In the world of Tooth and Claw 'predator' and 'prey' characters live alongside one another. Bird-people and rodent-people for instance.
In Gold Digger, a minor character who is half-unicorn is momentarily worried about helping someone get past a pair of dragons who may try to eat him. Thing is, not only is he currently dating a dragon (their present queen, no less), but his other "half" is, you guessed it, Dragon. Clearly, somebody played a little TOO much with their food.
In The Phantom The Phantom has his own little nature preserve in the jungle where all the animals live harmoniously, by all the predators being taught to catch and eat fish instead.