Section 97 of the UK Copyright, Designs and Patents Act 1988 is very similar to the Australian provision I quoted earlier. Once again, I have not looked at any case law. This is my uninformed opinion only. It is not legal advice and you should not rely upon it. When you download a pirate work you make an infringing copy. If you did not know and had no reason to believe that "copyright still subsisted" then no damages can be awarded against you. The other remedies it refers to may be injunctive relief to force the handing over of infringing copies, or accounting for any profits you may have made from the work. The latter would of course not usually be applicable to a simple download for private use.
It does not impose an absolute duty on the part of the person downloading to research the Copyright status. What constitutes a reason to believe which would deny a downloader the benefit of this defence? It is not defined, so will presumably be restricted to reasonable reasons to believe. So, if you see the Harry Potter books or the latest movies or music which reasonable people would know are not available for free, this would in my opinion give you reason to believe copyright may still subsist. In these types of circumstances certainly you should check if you want to avoid any possibility of being sued.
I think the suggestion that you are required to check if a work within the public domain in other countries is still subject to copyright in your country is simply wrong unless there is something else which indicates it may not be, for instance a notice on the Download Page for each book giving the relevant details for that book and a warning, possibly including the duration of copyrights in at least major countries. Or, even better, asking you to enter your country and refusing to download if copyright subsists in your country. Much less than this I doubt would constitute a "reason to believe". The hypothetical reasonable man on the Clapham Omnibus is not an expert on Copyright Law.
There also seems to be some suggestion that you have a duty to check everything before downloading because there is allegedly some pirated material on the site and allegedly the amount of such material is not totally insignificant. Once again, the site is called archive.org, is a non-profit with laudable goals in the Public Interest. A reasonable person would not suspect significant pirate content. Even if this is not so, it does not in my opinion mean that you have an obligation to check copyright on every download. Contrast, for instance, the Pirate Bay. The very name of the site may well impose such an obligation.
This provision is simply aiming for the application of some common sense. If only the rest of the Intellectual Property Laws were the same.
The relevant sub-section of the UK law is set out below for your convenience:
97 Provisions as to damages in infringement action.
(1)Where in an action for infringement of copyright it is shown that at the time of the infringement the defendant did not know, and had no reason to believe, that copyright subsisted in the work to which the action relates, the plaintiff is not entitled to damages against him, but without prejudice to any other remedy.