Tim, the handle resolution (redirect handle to the registered URL) is done by the handle service provider. In the case of DOIs that is CNRI, which provides a scalable distributed infrastructure able to handle billions of resolutions. CNRI started the handle system, but there are now alternative providers, coordinating via the DONA Foundation.
The organization that registers handles only has to worry about registration of handles and updates of the URLs the handles point to. This is done in the central handle infrastructure. This can be a lot of effort, made a little bit easier when using template handles. Related discussion at Billions of records.