Not necessarily, depending on what we're talking about. All animals have self-preservation instincts (or are supposed to, anyway) which helps them not die prematurely. Fear of things like heights and spiders are the way our instincts tell us, "Hey, this could possibly kill you, so could you stay away from it, please?" Generally speaking, we are naturally afraid of dying - but more specifically, of being killed.

Fear of dying naturally of old age, though, seems to be a purely human thing. We can't really know how other animals feel about such things, but from what anecdotal evidence I've seen, I'm inclined to believe that most animals are very accepting of their time to go. I think the human fear of natural death comes about from our desire to know everything, and we don't know what happens to "us" in a conscious sense after we die, and for a lot of people, that's scary.