Games were naturally a much more common application of a door. And since most boards only could handle one user at a time, most of the games were turn-based, one turn per user per day usually. Consequently, they drew people to systems, making them come every day to make their next turn lest they fall behind their fellow users. LegendOfTheRedDragon was one of the most popular. Some door games did allow real-time interaction with other users provided the system had multiple communication channels; indeed, this has become more popular as BBSes move to telnet (so to speak; BBSes are in decline).
These kinds of games can also be played over the Internet too, typically by e-mail. In the old days, chess by mail and the game of Empire were the most popular e-mail games around. Nowadays, increased bandwidth has allowed real-time games like EverQuest?, UltimaOnline?, and even Yahoo! chess to become more popular.
Consequently, DoorGames seem to be a thing of the past.
See also DistributedGame.
As HandHeld devices become popular, I predict a surge of turn-based games as maintaining a real-time environment in rapidly changing networks is extremely difficult. There will definitely be a place for game systems that allow people to synchronize in the morning to get the latest game model, decide your turn during the day, and then synchronize at night to make the change. It's really a cycle of technology, always moving backwards as we move forwards (but with better graphics). -- SunirShah