I used to shoot a rabbit almost every morning, and would feed it to a random dog.

Won't hurt them. None of the dogs ever ate the tail. All there would be left is a little "cotton ball"

Seriously though, you can't keep feeding rabbits to dogs; they don;'t have enough fat on them. You can feed them occasionally (so long as they don't have tularemia), but do not keep doing it.

The dog will starve / get malnourished; rabbits don't have enough fat on them.

Jack