Cory and Matthew
I have exactly the same problem: when the ATV is connected to my router wirelesslyand I'm using it to stream music via Airplay it will drop out after a time--not always the same. If it drops out, or when it goes to sleep overnight, there is no sign of the connection at the ATV end when I try to reconnect. Stopping and starting Home Sharing on iTunes and/or the ATV makes no difference; restarting iTunes or the PC makes no difference; only rebooting the ATV restores the connection. Then it will work just fine until it sleeps or Aiplay drops and around we go again.
I solved this problem early by connecting the ATV to my router by ethernet cable rather than by wireless. Now it always seems to pick up the connection when it has been asleep. Very occasionally--for instance when there's an update of the OS--I try the wireless thing again, but the problem recurs so I go back the ethernet connection. I'm lucky I can I suppose.
I've never had trouble streaming video over the internet to the ATV (say from the iTunes store); I did once have trouble streaming from iTunes on my computer--the program stopped after about 10 minutes and it took a while to restart it. But it was while I was testing the wireless connection. When I plugged in the ethernet cabnle it all worked happily.
I have also downgraded the OS on my ATV to 4.3 to get around an Airplay problem (needing to stop and restart Airplay to get any sound) documented in this thread https://discussions.apple.com/message/17212201#17212201.
Like Matthew I have also discovered problems with the remote app on both an iPhone and an iPad. It will work happily but then suddenly will not be able to find the connection. Restarting the app makes doesn't help; I can fix it only by restarting the iOS device. So, for example, last night I wanted to stream some video through an iPad app to the ATV, but it refused to see the ATV until I restarted the iPad. Then it worked flawlessly.
Go figure.
My layman's analysis of these posts is that there's really a whole range of problems with connectivity of the ATV being set out here--people are not all talking about the same problem. Although it does seem that the ATV will remain pretty happily connected to the internet in most circumstances and the issues and solutions here are about Home Sharing connectivity. I say again: I'm a layman, but that does tend to suggest a Home Sharing software problem in iTunes or the ATV.
Sadly for anyone who reads this, my relatively low level technical knowledge means the only way I'll find a solution is if I luck on it. But, given the apparent expertise of some of the posters, I retain an certain optimism that a solution will appear.
I've said this before, though: experts should not having to be searching for clever setting in the bowels of routers or software firewalls to make this thing work. Apple sells it on the basis that you can take it home, plug it into a power outlet, connect it to home sharing and, hey presto, it will work. There's 41 pages of posts here that say it doesn't. That's probably not quite good enough on Apple's part.
It seems to me the problem is exacerbated by Apple. We posters have no idea whether Apple is on to this; there may well be many earnest geeks in Cupertino tearing their hair out trying to find the bug and every bit as frustrated as we all are. Alternatively, as some of the unhappier posters here suggest, the said geeks may be cacking themselves laughing knowing it will never work and will soon be superceded. I'm a kind soul and would incline to the former view.
Even though the ethernet-connection-solution works for me, I'll keep following this post. It's impressive seeing how much time people are prepared to give up to finding and sharing solutions.