I have fic saved on my harddrive that will probably never see the shining lights of the internet again. It's stuff I enjoy re-reading in particular moods, much the same reason I keep books. I don't think an author has the right to tell people not to keep copies of something, and though I can think of a situation where I might think an author's request were reasonable ("I wrote that horribly racist/sexist/homophobic story before I understood that it hurts people" maybe? I've never liked a story like that, but I suppose it's possible someone might) I don't think people are obligated to delete something that's privately saved.

On the other hand, the author has every right to pull something down from public display IMO -- deleting their ff.net account, telling the Internet Wayback Machine to stop archiving their old web page, or even complaining to a webhost about a site that's archiving their work against their will. The difference to me is that fic on a website can still be found, while what I have on my harddrive can't.