Thanks to Tim Dunham for putting this together.
Instructions on how to add or remove an email account from Facebook. Use these instructions if you want to remove the address completely from the choices (you will still have the Facebook address, it just won't be displayed on screens that others can see):
Instructions on how to hide the Facebook email address on the timeline:
1. Go to the page that shows your timeline.
2. Look at the top center of the page for the "Update Info" button and click that button.
3. Find the "Contact Info" area at the lower right of the screen. Click the "Edit" button in the "Contact Info" box.
4. At the top of the screen that appears, there is a list of your email addresses. On each entry, there is a pulldown that allows you to "Hide From Timeline" or "Show On Timeline". Make the selections as appropriate.
5. Scroll to the bottom of this window and click the "Save" button to save your changes.
If you do not use the timeline, the process is similar if you wish to remove the address from your Profile.