Did you buy the map pack on that Xbox or did you transfer it from somewhere else?
DLC on the Xbox 360 is protected by DRM to prevent people from buying the packs on one Xbox, then transferring it to their friends. The DLC can only be played on the Xbox that originally downloaded the content. If you are on a different console, then you have to be logged in to use the downloaded content.
If there is a license problem, that could also cause these issues.
I remember hearing about a way to transfer the licenses to a new console, but I don't remember what it is, call Microsoft to find out how to do that.