Who originally coined the phrase, "Publish or perish," which is often
used to describe what some academics must do in order to keep their
positions at academic institutions?

It appears that the answer to your question is that the question
cannot be answered in a totally definitive way.

"Publish or Perish
What's the origin of this phrase? Our patrons want to know, but the
origin has been lost to the ages!"

Internet Public Library
http://www.ipl.org/div/farq/puborperFARQ.html

I think it's safe to say that the communications theorist and media
sage Marshall McLuhan popularized the phrase "publish or perish," and
McLuhan is probably largely responsible for the wide use of the phrase
in academia, but the phrase predates McLuhan. Please read this
document for details:

Homepage of Eugene Garfield, University of Pennsylvania
http://www.garfield.library.upenn.edu/commentaries/tsv10(12)p11y19960610.pdf

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Logan Wilson, a former President of the University of Texas, is
crediting with being the first person to use the phrase "publish or
perish" in print. It is still not certain whether he created the
phrase, or whether he had a source. Since Wilson is dead, the true
origins of the phrase retain a certain element of mystery.

"The phrase 'publish or perish' was apparently coined, or at least
publicized, by Logan Wilson in his classic study The Academic Man: A
Study in the Sociology of a Profession, 197 (1942) ('The prevailing
pragmatism forced upon the academic group is that one must write
something and get it into print. Situation imperatives dictate a
'publish or perish' credo within the ranks.')."

Homepage of Bernard J. Hibbitts, University of Pittsburgh School of Law
http://www.law.pitt.edu/hibbitts/lw_fn.htm 

Google search strategy:

Google Web Search: "the phrase publish or perish"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22the+phrase+publish+or+perish
Google Web Search: "publish or perish" + "first used"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22publish+or+perish%22+%22first+used
Google Web Search: "origin" + "publish or perish"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=origin+%22publish+or+perish
Google Web Search: "coined" + "publish or perish"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=coined+%22publish+or+perish

I hope this information is helpful. If anything is unclear, or if a
link does not function, please request clarification; I'll be glad to
offer further assistance before you rate my answer.

Best regards,
pinkfreud

It's not the precise answer, but I'm convinced it's the best answer
available anywhere. This is my first time using Google Answers and I
will be using it again. Many Thanks!

I notice that the link to Eugene Garfield's homepage did not turn out
well in my text above. For those who are interested in reading the pdf
file that I mentioned, the link above can be "cut & pasted", or you
may try this link:

http://tinyurl.com/2adlg

I apologize for the inconvenience. 

~pinkfreud