I have had the same issues trying to airplay from the eyetv app (both with live TV and with recorded shows). I've had the airplay button available, but the show just keeps playing on the iphone instead of the TV; I've had the show play on the tv but with no sound; I've had live TV work, but not recordings, and vice versa. The repeatable fix for me is to not just kill and restart the app on the phone (tried that, didn't help), but to actually reboot the phone. Usually this makes it work fine until the next time I try it out. . . usually a day or two later.
What's more, when I do get it to play on the apple tv, sometimes I can pause/rewind/ff, but most of the time I cannot. And pausing/rewinding causes the airplay session to fail. Hence, even when it does work, it's not a good example of DVR-lik experience, since I can't reliably pause and rewind a show I'm watching. (The pause and rewind functions usually work for me when watching on the phone itself, but not usually during airplay.)
I think the long and short of it is that the eyetv app is just plain buggy. Annoying since I paid $5 for it.
There are several alternatives to stream your eyetv recorded shows to apple tv 2g: you can have eyetv automatically export the recordings to iTunes, and then stream them to ATV2 via Home Sharing. If you jailbreak your ATV2, you can use XBMC or Plex to stream the eyetv recordings without having to transcode to iTunes. And if you go whole-hog and jailbreak the ATV2 AND purchase Firecore's ATV Flash (black), you can use the Couch Surfer Pro web browser to access your EyeTV via the EyeTV live 3G functionality, and watch live TV or recordings on the apple tv that way.