I would pipe all incoming mail on that mailbox into a PHP script that gets invoked (rather than simply placing the message into the mailbox container as would normally be done). Instructions vary on how to do this based on which mail daemon software is being used.
Alternatively, if the above method is not available, you could instead setup a cron job that periodically polls a given mailbox (e.g. via POP3 or IMAP) in order to retrieve and process all messages in it.