Chimpanzees specifically eat other monkeys as a form of nutrition. This doesn't happen in all species, but many species are pure herbivores but will still kill other species to defend their territory.

Lions don't eat tigers, because they will never get close enough to fight. During the Roman Colosseum they actually had to chain lions and tigers close enough to each other that they were both so uncomfortable that they would attack each other. The reason they both don't attack is because they invariably both die. Even Rome's celebrity lion, it's one of the few actual lions named by scholars at the time, died during mid fight due to so many injuries that a juvenile leopard, (IIRC) which is the smallest of all the big cats, easily killed it during its final fight.

Lions and tigers will never fight, they don't even exist in the same territory. Jaguar's live not only on a different continent, but in an entirely different hemisphere. That leaves the Leopard, which is the smallest of all the Big Cats and as such will always retreat from a confrontation.

However, Hyenas have been known to kill Lions and vice versa. In fact, out of all apex predator species the two behave strangely because not only do they guard their territory from each other, which is strange because apex predators usually disregard each other as it's deadly not to. However, Hyena's are occasionally the prey of a Leopard despite Hyena's being able to scare a Leopard away from prey.

It's exceedingly simplistic to assume something from animals given the extremely complex relationships and higherarchies that are carried out. Tigers are the most territory dominant of all the Big Cats, yet they have been seen to offer food to multiple females and cubs, they've even been seen offering food to completely unrelated males despite the fact they will usually fight over a prey.

It's somewhat true that an animal will not eat its own kind, but that's rather ignorant thinking because we're assuming a tiger is actually edible to another tiger. Humans aren't truly edible to another human, we're actually rather nutritionally poor by all standards. We suck at producing vitamins, your average goat produces 15,000mg of vitamin C a day, we produce 0mg in a lifetime. Eating a goat gets me nutrition, where as eating a human gets me rather nutrient poor protein.

Humans frequently have to be careful over what organs of animals we eat, because when we encountered Polar Bears people were getting Vitamin A poisoning. This not only likely happened when the Inuit started hunting Polar Bear, but is documented to have happened again when people started sailing to the North Pole.

Humans likely ate Neanderthals, but likely only as often as we ate each other, which largely depends on famines. When early man ran out of food, instead of choosing each other they might have chosen the Neanderthals instead. However, humans still eat our own species up to this day, and during famines, so do many other apex predators.

Pigs frequently eat their own young, in fact it's the leading reason for infant death in the species. Chimpanzees too are known to eat their own young, and sometimes others young. This extends to all cat species, all wolf/dog/fox species, a fair amount of primates, humans, bears and also happens frequently in certain species of fish. It's also documented in Bottlenose Dolphin, however it doesn't seem documented in any other aquatic mammals.

Then there's the whole sexual cannibalism thing in many species of insect. There is also size-structured cannibalism which occurs in many insect species, but notably in some squirrels too where the large will frequently consume the young, not usually infants though.

Quite literally, there is nothing in our D.N.A. telling us not to eat our close relatives, in fact there's more conclusive evidence on what species do eat their own than what species don't eat their own.