Closing your FeedBurner Account
FeedBurner is a service that is associated with your Google account. When your Google account is permanently deleted, information associated with your FeedBurner account (such as burned feeds and MyBrand domains) will be automatically deleted after a grace period of 30 days. If you wish to delete your Google account, please follow these instructions.
If you want to stop using FeedBurner without deleting your Google account, you can simply delete the feeds in your FeedBurner account individually from the My Feeds page in the FeedBurner control panel (by clicking a feed title, and then Delete Feed... at the top of the screen), and deactivate any remaining services from the My Account page in the control panel.