[Microsoft][SQL Server Native Client 10.0][SQL Server]SHOWPLAN permission denied in database 'database'.
After a lot of digging, I managed to track down an obscure post that mentioned DTA runs statements as the user who originally executed the statement - in this case the site's application pool user. This user is locked down and does not have SHOWPLAN permission, even though I, as administrator do have permission.
The solution in the development database (where security is not a major concern) was to grant SHOWPLAN permission to the application pool user:
grant showplan to [domain\user]