I'm pretty sure humans are herbivores, and we've been doing fine on meat. Not as fine as we could without meat. There are just extra illnesses to watch out for like what was listed above: heart disease, kidney failure, cancer, strokes. These illnesses could occur on a natural diet as well, so if there are already precautions in place for these things, the animal should still live a happy life (maybe cut short by some years).
Animals eating diets that aren't suited for them is not as rare as you might think. As I stated above, it seems likely that the natural diet for humans is (or at one point was) herbivorous. We have had several mutations over the years that have assisted us in being able to eat meat with less adverse effects: http://www.wsj.com/articles/SB10001424052702303933704577533271378820202. So, admittedly, giving primates meat is probably worse than giving humans meat. But at one point humans were eating meat without those mutations, and the human race still managed to survive.
We also tend to feed domesticated carnivores, such as cats, grains. Cat genetics has not been studied as much as human genetics, so it is not certain whether cats have also evolved to better be able to handle a diet that is contradictory to their nature. However, either way, a species going against its natural diet seems to be able to sustain a reasonable, though perhaps not optimal, life.
The situation may be different if a species was subsisting solely on a diet that is contradictory to its nature. However, it has been done. There are reportedly human communities (such as possibly the Inuit) who survive or have survived on meat-only diets: http://www.popsci.com/science/article/2011-04/fyi-what-would-happen-if-i-ate-nothing-meat. Many people who feed their cats a vegan diet, with supplemented vitamins, report that their cats live longer.
If giving an animal food that contradicts their biology, it is important to alter in some way. Humans face a high risk of contracting a food-borne illness if meat is not cooked properly, and grains in cat food are ground and mixed with the food. If giving an animal only food that contradicts their biology, the food should be supplemented. Cats can go blind without taurine, which is an amino acid that is created by animals and not plants. Humans on a meat-only diet could get scurvy from lack of vitamin C (if eating cooked meat) and constipation from lack of fiber.