Once you try to log out/unlink from Roblox on Xbox, it looks like you get transferred back to the sign in UI, however the background music and sound effects no longer play until you actually restart the Roblox app.
- Go to the Settings on Roblox Xbox
- Select Account
- Select Unlinked and Yes to the warning
Roblox will now have no sound until you restart the app. Background music abruptly stops and any sound effects do not run until you restart Roblox.