In Catacombs, how do the remaining party members deal with a dead Hero's possession?
No.

The game rules, pg 11 (12 of the pdf) say:


  At this point, that hero is out of the game unless the remaining heroes reach Althea the Healer and resurrect the dead hero (see page 16). Due to this possibility, the dead hero retains control of any spells, gold coins and or items that he or she has collected. Dead heroes may not use their gold coins or offer them to other players in any special room until successfully resurrected.