I had exactly the same issue when accessing my iTunes library saved on my remote computer from Apple TV.
My remote computer connects to a dual band ASUS router via Wi-Fi. Apple TV is connected to my router via Ethernet cable.
Another computer directly connected to the same router via Ethernet cable plays just fine without disconnecting home sharing after one song (then iTunes on the remote computer needs to be closed and re-opened in order for homesharing to work again).
After troubleshooting for a long time I've noticed that the AirPlay icon is also not showing on remote computer beside the volume slider
> it's not an Apple TV issue, must be a router/Wi-Fi issue.
> In order to check if iTunes is OK on the remote computer I've connected it directly to the router via Ethernet cable and that solved it. Airplay icon showing and no song interuptions.
At the same time I shut down Windows Defender, Router and Windows firewall and assigned a static IP address to my computer and also forwarding ports in my router that iTunes needs.
http://support.apple.com/kb/HT2463
I now forward port 3689 and 5353.
Ultimately not 100% what made it work but after half a day playing with it connecting to my router via Ethernet seem to have done the trick.
Later on I've restarted Defender and also both Router and Windows firewall.
All working fine ever since via Wi-Fi and Airplay icon showing as well.