There have been plenty of surprises (pleasant and un) at the Tonys, but very few true "shocks".
As mentioned before, Two Gents over Follies in 1972
Also, Brian Bedford in 1971 over Gielgud, Richardson and Alec McCowan.
My personal biggest upset happened in about 1975, Best Actor in a Play with six nominees including Henry Fonda and Jim Dale although there was no clear favorite. Presenter Jean Stapleton opened the envelope and said "And the winner is..."(she read the contents, looked front, paused for effect and began again), "...and the winners ARE..." I immediately thought 'tie', but no.
There was a double nomination for Best Actor that year. Two South African actors played roles in two plays by Athol Fugard (who also received a single nomination for writing both plays) that were presented in rotating rep in a commercial engagement at the now-gone Edison Theatre and received one nomination (like the Side Show leads), and they pulled out a win.
I cannot exactly remember their names: Winston Ntshona and John Kani come close, but the plays are titled Sizwe Banzi Is Dead and The Island.
The audience reaction was genuine--surprise followed by unalloyed glee and sustained applause as if to say, "The best men won."
"If my life weren't funny, it would just be true. And that would be unacceptable."
--Carrie Fisher