The common popularity of the phrase "dwarf toss" can be traced back to a single Australian bar, the earliest reference to which that I can find online is: March 05, 1985 Chicago tribune under the headline "The Dwarf Toss" reports "the most unusual tavern competition has to be the one that was held recently in an Australian bar. You may have read about it." - it regrettably doesn't mention where we might have read about it, so my trail dead-ends there.
This event garnered no small amount of press, outrage, and protests, causing even the UN's Human Rights Committee to eventually weigh in against the practice... thus guaranteeing that the practice would become internationally recognized, rather than only in one small bar (and one small dwarf) in Australia.
I can however find precursors to suggest that the practice of hurling a dwarf as a projectile existed even before this point, even if only in the popular imagination:
From 1911, Good Housekeeping - Volume 52, "decide whether to hurl the dwarf through a window, or toss him down the stairway." (earliest mention I can find of the word "toss" wrt dwarves).
From 1893, Railway World, "As easily as the giant may throw the dwarf."
In the 1878 Encyclopaedia Britannica's entry on Dwarf, "The last dwarf retainer in a gentleman’s family was the one kept by Mr Beckford, the author of Vathek and builder of Fonthill. He was rather too big to be flung from one guest to another, as used to be the custom at dinners in earlier days when a dwarf was a “necessity” for every noble family."
Counterpunch.org traces this claim to "E.J Woods, author of the useful “Giants and Dwarfs” (1860)" - however, since I cannot by searching find anything even remotely approaching that phrase in the 1868 printing of the book, I can't confirm.
Reading a short biography of Beckford, he lived from around 1760 to 1844. However, I am unsure if this is not just one more myth intended to blacken his name, taken as fact by the encyclopaedists. Elsewhere I find references that the man "shared the pornography", and "was accused of improper practices" with the dwarf, libels for which he successfully sued for damages - sounds like, when you're the son of the richest man in England, people like to talk smack about you.
Either way, dwarf-flinging has been written about at least as far back as the 1878 Britannica, and possibly practiced sometime in the 1700s if not before.