The earliest match I could find for the expression "assume room temperature" as a euphemism for "die" does indeed involve Rush Limbaugh. From Jeffrey Yorke, "WGMS Transmitter Sold," in the Washington [D.C.] Post (June 27, 1989):
Last week he identified himself to listeners as "Rush 'Warmonger' Limbaugh," and then applauded six schools in Pasadena, Tex., that prohibited students from wearing peace symbols because school officials believed the symbols were the mark of Devil worshipers. Later in the same program, a listener asked Limbaugh if he was sorry to hear of the death of liberal journalist I.F. Stone. "I'm not sorry at all I.F. Stone assumed room temperature," he said.
The next-earliest instance (probably) is from Magnetic North Magazine (1989–1990?):
Now that Emperor Hirohito has assumed room temperature, the N[ew] H[ampshire] Fish and Game Department is encouraging legislators to pass a bill that would stem the flow of illegal bear gall bladders to the Far East.
Hirohito died on January 7, 1989, so it is possible that this instance appeared before the Washington Post instance noted above, but the collection of magazines cited here runs across three volumes (presumably extending from 1988 to 1990), so the window is rather large.
The term next appears in Google Books search results in a letter to the editor of the New York Times Magazine from a reader in Suffield, Connecticut, in 1991:
Limbaugh's targets are the homeless and their advocates, minorities and the "NOW crowd." This is hardly a satirical hit list with which a modern-day Swift or Twain could identify. Nevertheless, a funny, right-wing conservative is a novelty that bears listening to—for a while anyway. I'll predict his act has another year before it Assumes Room Temperature (as Rush would say).
Limbaugh's book, The Way Things Ought to Be, mentioned in the Urban Dictionary entry quoted in BiscuitBoy's question, was published in 1992. For the next decade, the expression seems to have been especially popular in the Limbaugh-attuned right-wing commentariat. Examples appear in The American Spectator (1996), The American Spectator again (1997), The American Spectator again (1999), Charlie Tyrrell, The Rules of Power: Reclaim Your America from Washington Hypocrites, Liars and the Compassion Con Game (2000), The Limbaugh Letter (2001), and The American Spectator again (2003).
But instances also appear in John Kolness, The Absolute Last Chance Diet Book (1991), Gary Stubblefield, Inside the U.S. Navy Seals (1995), Robert Waring, Totally Unauthorized Guide to Doom II (1995), Steve Brown, Approaching God: How to Pray (1996), Mike Mayo, Videohound's Video Premieres: The Only Guide to Video Originals and Limited Releases (1997), Don Pfarrer, Guerrilla Persuasion: Mastering the Art of Effective and Winning Business Presentations (1998), J. Douglas Arnold & Zach Meston, Nintendo 64 Survival Guide, volume 2 (1998), and J. Lee Butts, Texas Bad Boys: Gamblers, Gunfighters, and Grifters (2001) ["Whether done with a pistol, rifle, derringer, or slingshot, George Wall assumed room temperature a day or so after the smoke cleared."—phrase not shown in snippet window].
The slang euphemism "assume room temperature" for "die" seems to owe whatever popularity it has primarily to Rush Limbaugh, although R. Emmett Tyrrell, Jr.'s American Spectator vigorously promoted its use as well. Dan Bron's answer clearly accounts for the sense of the expression.
Outside the satirical conservative milieu, the term doesn't seem to have especially wide currency, though the array of sources that have employed it—from videogame guides to a religious tract to a diet book to a business success strategy—indicates at least some seepage in multiple directions.