If I understand your question correctly, it sounds to me like you need two separate scripts: one to listen for incoming messages and log them and another to display the logs via http.

To listen for incoming messages, you need a daemon that listens to the port and logs messages to a file. You also mention that you want the ability to start and stop the logging from the web. One possiblity for starting the daemon is by forking a process...see forking from web for some ideas. Maybe have a "last accessed via the web" timestamp in a file that the daemon checks occasionally and deletes the file and kills itself when the time is not recent enough.

The web page would start the daemon, if not already started, parse the daemon's log file and display it as HTML. To refresh the page, use some JavaScript, HTTP refresh, or a META refresh (see Refreshing a Web page via Perl for some info).