Felynes are very cute, innocent cats who keep to themselves in the maps. You'll find most of them in the hidden Felyne Village that's in every map dancing around a campfire but if you do find them roaming about, they honestly aren't going to harm you and so you shouldn't harm them. Should a monster come along, they'll flee. If provoked, they'll sometimes whip out barrel bombs on you. Careful!
Interestingly enough, Felynes and their cousin Melynxes can never die. Instead, they'll just dig away so that they can live another day. They have a chance of dropping a shiny in the process. Most of the time it's account items or a supply box item.