Your Palico can faint, but the only detriment to that is you won't have their support for a short time while they recover. When it's health get's low enough, it will try to chug a potion on it's own before it does get knocked out.
A Palico is built a bit different than a player, as in.. they are a monster. They have a much higher healthpool than a player(you can see this by dropping a health pod down and seeing how much slower their health gains compared to yours), and as they level up they get even more. Being a tiny creature also means their hitbox is also small... they won't get hit too often as they scamper around.
There is no penalty to your hunt rewards on your palico's performance. That said, keeping them alive can be very beneficial as their gadgets can be very useful to you, so ignoring their equipment isn't a wise thing to do.
Now if you have found a Tailrider in your travels and they end up fainting, they will not recover and will no longer assist you until you go grab another.