Instructions
Turn on your Xbox 360.
Open the "System Settings" dashboard menu option.
Open the "Memory" category. Select "All Devices" from the list of options on screen. Select "Gamer Profiles" from the list of options on screen.
Highlight the gamertag you want to remove and select the "Delete" option. Choose "Delete Profile Only." This will remove only the gamertag from your Xbox 360 and will leave all of your various achievements in place on the console.