This is a curious claim that I cannot substantiate. After looking around, I see the same claim in Wikipedia's article for Copyright renewal in the United States, but the cited source (Fishman, Stephen; 2010. The Public Domain: How to Find & Use Copyright-Free Writings, Music, Art & More) doesn't appear to provide a robust citation. (On page 287 in Chapter 15, Fishman writes, "The Copyright Office estimates that only about 15% of pre-1964 published works were ever renewed", unaccompanied by a footnote.) One also notices that 1961 and 1964 are not the same year.
I stumble across a similar claim ("a 1961 report from the U.S. Copyright Office estimates that 85% of the books never had the copyrights renewed") on lcgsco.org, which disappointingly turns out to be the website for "Larimer County Genealogical Society", with no link to the report in question.
The Copyright Office does make available its historical archive of annual reports at https://www.copyright.gov/history/annual_reports.html. Could its annual report be the one referred to? Searching for the string "renewal" in the 1961 annual report turns up two noteworthy occurrences (out of a reported nine):
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on page 2, a claim related to renewals mentioning a "15 percent" figure
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on page 16, a description of "Studies 29–31 [… including …] 31. Renewal of Copyright"
The full sentence on page 2 where the "15 percent" figure appears is actually that "The year's increase in registrations was nearly 3 percent, this was counting a 15 percent decrease in renewal registrations, the result of the corresponding decrease in original regisrations 28 years previously." This is, troublingly, not the same thing as fewer than 15% of copyright registrations until 1961 being renewed. (It is not even the same as only 15% of registrations up for renewal in 1961 being renewed.) It is a 15% decrease in renewals relative to the prior year, i.e., number of renewals in 1961 compared to the number of renewals in 1960. If this is the report that the is meant to be the source for those claims of 85% non-renewal, it then it fails verification.
I have also come across a reference for renewal figures in a 2008 article in D-Lib Magazine (Peter B. Hirtle, "Copyright Renewal, Copyright Restoration, and the Difficulty of Determining Copyright Status"). Footnote 4 there cites the same Study 31 (attributed to a Barbara Ringer). Interestingly, it is not cited as a way to substantiate the 15% claim, but instead that "only 7% of registered copyrighted books" were renewed. It's not clear from context whether that claim is meant to be about the full range of potential renewals up to 1961, or merely the subset of works up for renewal in 1961.
At the time of this comment, I have not yet looked into Study 31 itself. I expect it to contain answers.