At a guess, and this is a guess… the cigarette will burn through the tissue of the eyeball itself. This would probably manifest as a massive fluid discharge, as the eyeball ruptures and collapses.
This will destroy the eye in a matter of seconds. Holding it there for a minute is both overkill and much harder than it sounds.
Medical attention probably isn’t going to be able do anything. Not immediately after the injury; not an hour later. The eyeball itself would be destroyed, that character’s going to be blinded permanently.