All hail, by the way, IMDb's much-improved search-by-year function! Which makes this, if not more possible, then certainly much quicker.
BEST SUPPORTING ACTOR, 1986
Tom Berenger, Platoon
*Michael Caine, Hannah and Her Sisters
Willem Dafoe, Platoon
Denholm Elliott, A Room with a View
Dennis Hopper, Hoosiers
Julian Beck, Poltergeist II: The Other Side
Michael Caine, Hannah and Her Sisters
Dennis Hopper, Blue Velvet
Ray Liotta, Something Wild
Tom Noonan, Manhunter
With an honourable mention to Jeffrey Jones in Ferris Bueller's Day Off. I consider Daniel Day-Lewis a lead in My Beautiful Laundrette -- though not, of course, in A Room with a View. Chime in, friends! I think this one could run and run.