Nothing, it ignores your request. It doesn't ignore it when DNT is disabled. As Piwik ignores DNT requests by itself, I guess nobody bothered to disable the snippet on server side when DNT is detected.
That's a shame. Although I guess it is difficult to avoid the request if you have lots of static pages with the script link in them.
Perhaps we could have something like a <link> or <img> track flag, so my browser could decide whether or not to fetch the link based upon DNT settings?
They know that someone accessed resource X from IP address Y from browser Z with language preferences A, encoding preferences B, SSL cipher suites C (when https is used), DNT preferences D, sometimes also protocol support E (when upgrading to SPDY or HTTP/2.0), having a TCP/IP fingerprint of OS F. I guess I still haven't covered it all.
Exactly, that's the sole purpose and that's how DNT works. It's nothing more than a voice "I don't wish to be tracked" and an expectation to be listened to.