Although the implementation is limited to running from the desktop only, YouTube live streaming has now come to the Chromecast. It's interesting, because even though the Android app still reports an incompatibility with YouTube live streams, almost every live stream is actually working on the Chromecast.
In fact, there is reason to believe that this isn't quite official yet, because Google hasn't updated their support page to remove the limitation as it did for when they enabled private videos recently. Here's a quote with a few more details,
You can activate a live stream on the Chromecast just like on any other video – click the button and set the playback location to the dongle rather than your computer. It's not clear when Google will expand this to the Android app, seeing as it clearly works now. Maybe it'll take an app update this time.
Thanks to the folks at AndroidPolice for sharing this info.