Is it considered an ongoing sin for a person if he possessed things containing unlawful elements and used them in a permissible and sound manner (such as muting music in a game or avoiding bad scenes within it), and then his heirs misused them after his death?

If it is possible to mute the sound and hide the forbidden scenes in the game, then there is no harm in using it. There is no sin upon the owner of the game, Islamically, if the heirs use it in impermissible ways; the burden of sin is upon them. It is recommended to raise children upon religious values and upright conduct.