This must have changed because when I tested it when I wrote the post above I created the game, dropped a potion and left immediately. I waited 4 mins and took another char in, quit, took another in 5 mins after that and the game was gone.In regular Diablo, if you create a game and leave it after less than five minutes, it persists for 30 or 60 seconds. If the game is older than 5 minutes, it persists for 5 minutes. I'm not sure if my numbers are entirely correct, however.
BTW, if I remember correctly, when dropping a quest item in a game of a different difficulty, in particular a lower one where you have solved the quest already, you might not be able to pick it up again, so you might have to mule it in a game of the same difficulty as the quest's.