I have double checked setting and have no issues logging in with Remote Desktop. I can access all controls and use the computer fully. When I first log in for a moment the audio drops and then no issues but once I end the session and select disconnect. Once I hit disconnect Playit live app ends. I am running SHOUTCAST server and Winamp. Both remain working as does chrome, IE and file explorer. I never had issues with this when running windows 7?
I had the same problem earlier, never really figured out why. PIL just crashed everytime I closed Remote Desktop on Windows 10 and in the Event Viewer I could see that the process had been terminated due to an unhandled exception, I think it was with from the bass.dll module. Seemed like a driver issue or something. Perhaps something related to Virtual Audio Cable (that I use so get the sound from PIL to audio processing software + icecast).
I could sometimes keep it running by killing the remote desktop process using task manager instead, but ultimately I decided to downgrade the machine to Windows 8 (the oldest version that machine could run) and after that I've been having zero issues. There was also another issue with RDP and Windows 10, that when PIL didn't crash on exit the sound was stuttering for maybe a minute after closing. This issue also went away using Windows 8.
Not an ideal solution to run an older Windows version of course, but I found no other way to keep it stable. If anyone has found a solution to this issue on Windows 10, I'd be happy to upgrade again.
Please try using the Play on remote computer option when logging in via remote desktop (see http://support.playitsoftware.com/support/solutions/articles/5000620291-audio-does-not-play-or-record-when-using-remote-desktop-rdp-)
When you log in via RDP by default it replaces the audio driver with one that redirects the audio back through the remote session. When you log out, I assume it tries to reverse this.
If this doesn't work, I recommend using a different remote session software, such as TeamViewer.