Well, it varies between episodes. Rosebud has Mr. Burns ditching his parents for his grandpa. The Mansion Family indicates that Mr. Burns murdered his parents or at least committed manslaughter against them due to them "being in his way". Homer the Smithers indicated that his mother survived an attempted assassination of her by her son pulling the plug and was alive up to the present day (with the motive being out of revenge for her dating Taft), and Bobby, It's Cold Outside indicated that Mr. Burns' parents had died while Mr. Burns was put into a school, and their ghosts fading away due to being too late to embrace them.
Basically touched high-voltage wires and electrocuted himself to death after being driven insane by Springfield's dysfunctional state, especially after his attempt at exposing Homer as a complete idiot backfired spectacularly.
Died from a shirt launching accident during a freeway race (which Homer had been indirectly responsible for both due to ducking at the worst time possible and his parking in the ambulance reserve spot.).