So how exactly does this work if someone already bought it as a Community game? Do you have to pay for it again when it gets "promoted" to Live Arcade? Does the Community version get removed from the marketplace?
You're getting confused. There is no community game version of The Dishwasher.
Over a year ago there was a brief demonstration of Community Games that were in the Dream.Build.Play competition on Live, one of those demos were of a pre-build of The Dishwasher. At that time there was no Community Games service. The winner of the contest would get the chance to publish their game as a full Xbox Live Arcade title.
Since then the developer has been getting it ready for XBLA.