Question: My cat catches a lot of mice and birds. It always leaves the heart, perfectly intact. My question is, what about the tails of the mice, the feathers of the birds? I sometimes find some feathers but never all of them. Does the cat eat it, and why does it leave the heart?

Answer: 
Cats remove the feathers by licking them, and they’ll inevitably swallow some of them in the process; their mouth structure doesn’t really allow spitting stuff out like ours does.

If it’s “your” cat, I assume you feed them? If so, they hunt primarily because their instincts tell them to, not for nutrition. They may eat the tastiest/easiest parts of their prey, but they’ll leave the less tasty or more difficult parts. Feral or stray cats will usually eat the entire prey, including bones, because they *need* to.
Maybe the hearts are too tough to eat even with their sharp teeth, or maybe they are actually poisonous and therefore inedible. If either is the case, the cat knows, and therefore leaves it alone. As for the feathers, I think they are discarded since they're most likely as useless to the cat as the hearts.