nzgeek: The problem isn't related to HTML 5, it's with the site itself. I've just tried the site in Chrome for Android, and it still doesn't work. The latest version of Chrome for Android scores 490 out of 555 in the official HTML 5 test, which is higher than both the latest versions of Internet Explorer (Windows) and Safari (Mac OS X).
The error message says: This player is not enabled for HTML5 delivery. This wording says to me that the player software they're using is capable of HTML 5 playback, but hadn't been configured to allow it. Chances are that TVNZ's licensing arrangements require then to use DRM, but they don't have the necessary infrastructure in play to work with HTML 5's DRM support.
Your best bet is to sideload the TVNZ app. Download links: phone, tablet. You'll need to allow sideloaded apps to be installed (Settings > Security > Unknown sources) but this option can be changed back immediately after installation.
The stupid thing is that while the app itself is completely free and available on the Play Store, only Samsung devices are allowed to install it. Others have installed the app just fine on non-Samsung devices, so the restriction is completely unnecessary.