Overlay journals are one of the most interesting experiments I've seen in open access publishing. An overlay journal relies on existing third-party document repositories to hold its contents. For example, several journals have been started as overlays on the arXiv: papers are published in the journal by placing them on arXiv, with the journal pointing to their arXiv URL. The main benefit of overlay publishing is, of course, that the journal does not have to maintain the document or its storage.
There have been several overlay journals, but they have mostly failed. The (immediate) reason is that libraries do not need to subscribe to journals whose contents are available, for free, on the Internet. For example, Scott Morrison says:
My limited understanding (having spoken to Greg Kuperberg yesterday) is simply that being an arXiv overlay journal was financial suicide. Annals found that libraries were dropping their subscriptions, and simply could not be persuaded to continue forking over cash when all the content was guaranteed to be available online for free! (I think this is a very important point for open access advocates to keep in mind—funding a journal through library subscriptions is flatly incompatible with open access.)
Morrison makes an excellent observation that is worth emphasizing: funding a journal through library subscriptions is flatly incompatible with open access.
That said, I think Morrison errs by associating overlay journals with this problem: it is a problem for any open access journal, not just an overlay journal. And it obscures the tremendous contribution that libraries make to journals, above and beyond subscription income.
Consider paper journals (which libraries still support). Not only do libraries pay for subscriptions, they store paper journals in expensive, environmentally-controlled, continuously-monitored facilities, ensuring that the contents of the journal will survive even if the publisher's copies are destroyed. And libraries are the only way that most people can read paper journals at all.
In the case of an electronic arXiv overlay journal, libraries are still providing these services. The arXiv is maintained by a library (the Cornell University Library), which runs the web site and backs up the contents. Moreover, the arXiv is mirrored by other libraries, providing additional redundancy and availability. This service is provided very efficiently, at a cost of around $7 per submission, but in any case, it is paid for by the libraries, not the publishers.
I don't think most publishers fully appreciate what libraries are doing for them. Even the publishers of overlay journals typically make the mistake of storing only their articles in the third-party repository; they still maintain a website for metadata (like the table of contents of each issue), as well as additional services like search. Overlay publishers should instead eliminate their web sites and place all content, including metadata, on the third-party repository. Services like search can also be handled by third parties. This would be both cheaper and more archival (there are plenty of examples of electronic journals that have gone out of business, taking their web sites with them).
Libraries are facing their own existential crisis, just as publishers are—their traditional functions of distribution, access, indexing, and research are in danger. They would like to continue their significant contributions to the publishing ecosystem, and publishers should view them as allies, not enemies.