davidator, you inspired me to get off my butt and create the example to support this feature. So here is the first pass at it. This will be included in the next version of the software.
Your website needs to be able to support PHP (I can't imagine a hosting service that doesn't). As davidator demonstrated, there are basically two pages: (1) the "listener" page that the TD sends the data to, and (2) the "viewer" page that one can point their browser at and see live information from the tournament. I've divided the viewer page out into HTML, CSS, and Javascript files. This should make it very easy to adapt and extend.
Attached is a ZIP file containing 4 files:
statusListener.php
This is the "listener" page. Once installed on your site, you'll need to configure the Status Updates feature in the TD to point to this page.
TournamentStatus.html
This is the "viewer" page. Once installed on your site, point your browser to it to see the current tournament status.
TournamentStatus.css
The stylesheet for the viewer page.
TournamentStatus.js
The Javascript supporting the viewer page.
The listener page will need to be able to write a file locally. You may have to configure your web site to allow this. You may need to modify the URLStatus variable in TournamentStatus.js, depending on where you install statusListener.php and TournamentStatus.html. By default, if the two are installed in the same folder on your web site, it should work out of the box.
The page automatically updates itself, so once you point your browser to it, you don't have to do anything else. The page will continually refresh. The page uses the JQuery Javascript library to make the values fade in and out (among other things). I did this to convey the idea that the page isn't "real-time" and to make it obvious when something on the page changes. The TournamentStatus.js file has some options at the top you can change yourself. The Javascript is basically self-contained, so it shouldn't interfere with any other script you may have on your site.
The code isn't written to handle odd situations, like a request to update the page taking a long time or failing entirely. So you may get strange behavior should something out of the ordinary happen.
The refresh rate by default is 3 seconds. This means that the page will request an update from the web server, and once that update is received, the page waits 3 seconds before issuing another update request. If the request from the server takes 3 seconds to arrive, the effective refresh rate will end up being 6 seconds.
On a local server, I set the TD to send updates once per second, and had the page refresh every 2 or 3 seconds and it has a nice effect. Might make a (potentially big) difference when updating to and viewing from an internet web site.