Is it possible to use AirPlay on a MacBook Pro 2013 with the lid closed?
You can prevent your MacBook from going to sleep with NoSleep.
The best thing you can do is turn the Light of the Screen down and use nosleep (or just disable Screensaver and Energysave mode).
On Macs closing the Lid does a hardware Trigger to get your mac to sleep. Don't mess with this because when the Lid is closed the Air-Slices at the Back are covered.
I know one can do this on Windows/Linux Notebooks but you shouldn't do it there either cause they will also overheat.