Thing is, if I plug my MBA into my 46" Sony TV with an HDMI adapter and watch a movie, the battery level drops much faster than watching it on the built-in screen. Even if I dim the built-in screen all the way down, the battery still runs down faster when using the external screen.
So more energy is being used to drive the second screen, and that might result in a warmer computer. However, I have never seen the fan kick in while doing this.