No sooner did we figure out Google added support for private YouTube videos to the Chromecast, than live videos started working as well. We've been checking around and it looks like almost every live stream is working on the Chromecast, but only from a desktop web browser right now. The Android app still reports an incompatibility with live streams.
Google hasn't updated the support page to remove this limitation like it did for private videos, but it's definitely working. If you want to test it out for yourself, try the Hummingbird Live Cam and the always weird QVC. This takes care of two categories of videos that were still not working with the Chromecast, leaving only non-mobile enabled videos as the lone holdout. This is a much less common thing to come across on YouTube, so maybe it won't matter much in the long run.
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.