Most authors would have no problems with that. However, the ones you need to worry about are the ones that DO have a problem with that, people who want to keep track of where their stuff is on the Internet. There's a simple way to find out: Ask them.
Most authors have contact information of some sort on their fanfiction.net Author Profile. The ones that don't, you can contact by leaving a review. If it's an unsigned review, be sure to include your e-mail address. If it's a signed review, they can use your contact information to contact you. (So, make sure that you have contact information on your own fanfiction.net Author Profile; if you don't have an Author Profile, you can make one, if you want - you don't have to actually write to have a fanfiction.net account).
If the author says "no", then don't archive their story. Most of the time they will say "yes". The frustrating cases are when the author doesn't say anything at all, which usually happens because the author's information on his account is out of date (i.e., they don't use that e-mail address anymore, or whatever), they aren't writing anymore, etc., etc. My advice to you in those instances would be to NOT archive their stories. Better safe than sorry.
Of course, nothing's stopping you from archiving whatever you want. From everything that I've heard, authors of fanfiction seeking actual legal protection never actually get any, so if they don't like what you've done, all they can do is fume, threaten, and flame you. I'm guessing you want to be a Good archivist, rather than an Evil archivist, so get permission before you archive anything!
Note, in the author's forewards of many stories, the author explicitly gives or denies permission to archive his or her works. What you should do in those instances should be obvious.
Author