The worn key is an unobtainable item intended to be used during Mourning's End Part I.
Presumably, the key would have been required in order to enter the Head mourner's office. For unknown reasons, this was removed, and players may simply enter the office as long as they are wearing full Mourner gear.
The key has never been obtainable in Old School RuneScape, but it remains in the game files.
|10 December 2015
(update | poll)
The examine of this key has been updated from "This key has seen a lot of use." to "This key has seen a lot of use (Used in Mourning's Ends Part I)." so its purpose is better described.