Erm, I don't think they would DIE die...more like, you would take a certain path in which a particular character is no longer able to be a part of your team or something? For example, "death" may come in the form of that device Wario has that turns characters instantly into trophies; he would use that against someone and kidnap them, somehow the trophy is destroyed, and for the rest of the course of SSE they are unusable. Or something like that.
Basically I think that's the closest thing to death in SSE is choosing a branch of the story that makes a character unusable. Such as the first scenario while fighting Petey, where you lose the princess you don't save (not taking into account that perhaps you'll find them later on, although that's pretty likely). Or maybe the old scenario of someone staying behind in a hazardous situation to save everyone(LOL).