I totally hear you! I'm running Mac 10.7.3 and EyeTV version 3.5.4 and I've been running into issues as well. Every once and a while I need to unplug the USB dongle and plug it back in to get the TV signal back. I've missed a few of my captured shows because of this.
I was running Mac OS 10.6.8 prior to this without issues. It's just when I upgraded to Lion.
Since I re-installed eyetv on my machine, I have a time shift between the audio and the video channel. Upgrading from 10.6.8 to 10.7 3 hoping to solve this issue I now have only still images with the audio channel runnig. This even happens when only playing a tv channel in time. Totally unusable. I already contacted the help desk, but got no answer. Anyone knows this?