Dying rooms are mentioned in Erich Maria Remarque's All Quiet On The Western Front (1929). After being wounded in a WWI battle, the protagonist recuperates in a Catholic hospital. He shares a room with several other wounded soldiers, and all of them dread being sent to the dying room. When the nuns come to move one of them to the dying room, he becomes hysterical and violent because the move signifies their loss of faith in his recovery.
I just saw that a couple weeks ago. Very disturbing and sad. Good movie though