You could do this in different ways.
Solution #1: PHP script or similar.
Have the script go to
http://site.com/logs/upload.php, with POST data:
User: user
Password: password
Date: <date that was logged>
Log: logfile here
Then the PHP script adds it to a MySQL Database, or writes it to a file named <date>
Then you could either access the log by
http://site.com/logs/log.php?date=date or by
http://.../logfile.txt.Solution #2: FTP
This is most likely the easiest one to implement.
Just have mIRC connect to the server via FTP, login etc, then upload the file through the FTP protocol. You would access the log though
http://.../log.txt or whatever you named the log.
I would recommend the FTP way, as it causes the least server load, however it isn't as configurable as the PHP method.