I hunt birds because I like working with dogs. If I ever get to where I don't care about the dogs anymore (not likely), I'll have to reassess whether it is worth it to hunt birds. Dove and quail, mostly, but sometimes pheasant, too.
Just love seeing a dog get on a bird. Don't even have to be hunting, or get a shot, if the dogs get to work.
I can't think of many things I won't shoot. I can think of things I will shoot that I won't eat, but there are other reasons than eating to have to do that, as discussed earlier. I don't see me going on a hunting safari anytime soon, so most of my hunting will be confined to the native and feral species populating my region of North America. Most of them, I'll have to confess, I'd shoot if the occasion warranted it, even though I haven't had that opportunity on any but deer, elk, quail, dove, pheasant, some other birds, rabbit, squirrel, coyote, rattlesnake, skunk (passed, thanks!), and a few others.
Why didn't I shoot the skunk? Didn't want to!