Can I move SQL Server backup files on a running database? Our backup drive is filling up with TRN and BAK files. I have administrative privileges on the server, but I don't have a login to the database.

Can I move the oldest files off to somewhere else without causing service interruption? Which ones?
In general, yes, you can move the backup files.  Backup files are not in use by SQL Server once the associated backup is complete.  .BAK files are typically full or differential backups, and .TRN are typically transaction log backups, and they can all be moved.  It is certainly possible to use different file extensions for the backups, but it's not common.

Please be aware, though, that there may be other processes that use the backup files, such as log shipping or automated restores to non-production environments.  There is, of course, no way we can tell you what may be happening in your environment, so we can't tell you what might break if you move files.  If you are just moving the oldest files, then it's less likely you'll be interrupting something.