"The frontier moves with the sun and pushes the Red Man of these wilderness forests in front of it until one day there will be nowhere left. Then our race will be no more, or be not us." — Chingachgook
Peak libraries is a concept named after the (contested) theory of peak oil. According to the peak oil theory, as the world's oil supply becomes depleted, its supply will diminish along a mathematical curve, causing large economic shifts and forcing alternative power supplies to be explored.
The peak libraries theory states that traditional libraries and librarians, such as those that have existed since the Library of Alexandria, are similarly doomed due to the advent of information technology.
Just as the unskilled assembly line worker was replaced by machines or the elevator operator was eliminated because the general public can use modern elevators directly, librarianship will cease to exist (or be radically altered) by the automation of library work -- e.g., as evidenced by the rudimentary service robots already in place in a few libraries      -- and the growth of patron-empowering technology which eliminates the need for librarians to serve as middlemen between library users and the services or information they need -- such as the existence of self-checkout systems and the potential development of truly intelligent searching computer systems, which would practically eliminate the need for reference question negotiation and other forms of library instruction.
 Other "Death of Libraries" Predictions
The end of libraries has been predicted before. Critics of the open stacks movement at the time feared that allowing patrons direct access to library materials would entail a decreased need for librarians. Instead it necessitated a greater need for librarians to educate the growing number of library users. The invention of microforms, the early and unrealistic "paperless office" crazes, and developments in ebooks and eink technology have also all caused some gloomy predictions about the future of libraries. So far all of these predictions have turned out to be very far off the mark.
The predictions failed because new technologies by themselves will not really change a profession. A contemporary doctor uses machinery that didn't exist a thousand years ago, for example, but the general role of a doctor as a healer remains the same. Likewise, the development of digital forms of information and their rising popularity over printed materials does not in and of itself entail the end of libraries. Indeed, just as with the opening of stacks, librarians are needed more than ever to help users navigate the complex and disparate information systems of today.
Let us look at the general role of libraries, then: they were started to give communities access to a broader set of information than possible as an individual; they act as intermediaries between publishers and patrons. Librarianship was grown out of the need to maintain discrete, physical collections and for librarians to train people how to access them.
As technology increases, however, there is more direct contact between publishers (or even authors) and consumers of information (cf. Google Scholar and Google Print), and a growing framework for an automated adaptability to user needs which has the potential to bypass the traditional role of a reference librarian via making libraries as easy to use as an elevator. Put more simply, who needs a librarian once you have Vox?
 Death of Digital Libraries
Predictions of the death of physical library assume the presence of an essential element: electricity. Without a cheap, abundant electricity supply, digital libraries would not exist. Herein lies the advantage of physical libraries: human energy is expended when opening a book, whereas fossil fuels are required to power the millions of servers and microcomputers that comprise the internet and by extension, digital libraries. The implication of peak oil theory for the digital library is simple: the peaking of oil and natural gas production threatens the production of electricity which in turn threatens everything electrically powered. Yet, blackouts or the threat of such are occuring with increasing frequency.  
 Criticism and Alternatives
Since we've yet to see anything remotely resembling artificial intelligence (as awkward results of machine translation and indexing illustrate), it could be argued that the technology required to create HAL or Vox will never exist, and therefore the need for human librarians will remain.
Libraries aren't just information warehouses. The information commons movement and the trend of placing coffee shops in libraries attempt to preserve the social role of libraries as communal spaces. Some libraries have also focused efforts on creating institutional repositories using modern publishing systems. But are these just stopgaps? Time will tell.
 Further Reading
- Libraries -- Future Bibliography from The Infography
- Susan E. Cleyle and Louise M. McGillis, Eds. Last One Out Turn Off the Lights: Is This the Future of American and Canadian Libraries? Scarecrow Press, 2005. ISBN: 081085192x.