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It isn't just a ban from Live, it also does this as well:

Why does my banned Xbox 360 no longer install to hard drive?
Once a console has been banned from Xbox LIVE, in order to protect the integrity of the Xbox ecosystem and its members, that console loses its ability to create trusted content.  This means that, for example, the console can no longer create profile and game data that is recognized by other Xbox 360 consoles or on the Xbox Live service, and it cannot create trusted game files or recognize previously created game files using the install to hard drive functionality.  The inability to create trusted content does not impact the console’s ability to play from discs in the optical disc drive.

http://www.xbox.com/en-US/consoleban