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I made a three-player game a while back, and the same things apply as to
any more-than-two-player game--the King need not respond to 'check' (it
may not be his TURN next,) the King may go 'suicidal,' (interpersonal
play ALWAYS causes weaker players to gang up on a strong one, and you
might trust the following player not to take advantage of your King's
exposed condition,) and the pieces of a 'dead' King must be dealt with
somehow.
It hadn't occurred to me to remove the dead King's pieces; I decided to
let them remain as obstacles, capturable at leisure as required by the
surviving players (on a real table it's fun to have a bigger stack of
'prisoners',) and actually your 'pandemonium' which ensues by removing
the dead King's pieces would be quite exciting, and in a way, for him, a
revenge from the grave. 'Kill ME, huh?!  Well, here's some confusion for
ya, ya dogs!' What fun. Good idea and thanks for it.