Are there any known Universities that refuse to pay for paywall access to academic journals?
No University has enough funding to spend on subscription fees for all existing paywalled research articles (e.g., Harvard University says it can't afford journal publishers' prices). As a result, some of their faculty and student resort to free alternatives SciHub/LibGen/emailing colleagues/etc.

Many universities located in developing countries cannot afford paying for paywalls.

If you need a particular University name, here are some:

University of Sierra Leone:


  When he needs new books to teach one of his courses, Professor Ibrahim Abdullah orders at least two from abroad: one for himself, one to give to the university library. If he needs scholarly articles, he writes to his friends overseas and asks them to send copies, since the university cannot afford journal subscriptions.


All public universities, research institutes and state agencies in Peru (translated by Google):


  Bad news for research and technological innovation come from Peru. The
  National Council for Science, Technology and Technological Innovation
  (Concytec), a government agency, will no longer offer free access to
  the ScienceDirect and Scorpus databases by the end of the year. The
  lack of funding from the central government has been the cause of the
  closure of these platforms for Peruvians.
  
  Access had been enabled for public universities, research institutes
  and state agencies. ScienceDirect (Freedom Collection) gathers more
  than 1,800 titles in full text scientific journals in 24 thematic
  areas and Scopus meets "relevant sources for basic research, applied
  research and technological innovation and is a tool for bibliometric
  studies" according to the portal Concytec .
  
  Access was open since 2014 and during that time 3.7 million full-text
  documents were downloaded. The downloads, according to Concytec, would
  have cost US $ 131 million if they had been isolated. While
  subscribing to these services for three years cost the agency only US
  $ 10 million.


A few other places:


Dutch universities start their Elsevier boycott plan
Germany-wide consortium of research libraries announce boycott of Elsevier journals over open access
Taiwan Tech to Discontinue Subscription to Elsevier ScienceDirect Starting 2017




Some interesting maps showing the location of Sci-Hub users (at least the location of the machine that makes the final request):