It’s possible to delete your account, but unfortunately it’s not very easy to do so.
Go to http://help.fitbit.com/customer/portal/emails/new and fill out the form (select something random from the product dropdown list) and request for account removal.
From their FAQ:
If you would like to cancel your Fitbit account you may do so by emailing our support. When you request cancellation of your account, your personally identifiable information will be deleted, including but not limited your email address, name, photo(s), friends list and links to social and partner sites such as Facebook Twitter, and Google Health.