Yes, they can feel pain, but just like humans a mortal wound will put them in shock, a state where they are dizzy, confused, and not feeling much at all. This is why its important to only take high percentage shots. Hit them in the vitals with an appropriate gun and it will be an ethical harvest, shoot their jaw off so they starve to death and they will die a horrible death. If you know and stay within your and your guns limitations you have nothing to be ashamed of, if you shoot at all brown you see or take long shots with a shotgun you have no business being in the woods.

"guns aren't for killing people, guns are for killing dangerous and delicious animals"
- Homer Simpson