I have a first generation MacBook Air that heats up a lot when playing videos, and almost grinds to a halt after a few minutes of playback.
Is there a way I could turn off the video during FaceTime calls so that the audio could continue smoothly?
All reports seem to indicate that there is no way to use facetime on the mac without video. You can continue a facetime call audio-only on iOS devices by pressing the home button, but oddly enough you can't disable video for the mac.
One suggestion I read awhile ago for low end machine was to place a piece of black paper over the camera. It takes much less processing power to encode a static image of mostly black than your face.
Minimizing the window will disable the video reception decoding, and may save you some processing power. Taping over your camera and minimizing the window might give you enough processing power back to use audio comfortably.
You might also consider skype or google voice if all you need is audio.
If you are having trouble with heat due to video processing through FaceTime, you should look at alternatives for your Mac. It is not just the video you are sending but also the video you are receiving that is causing your processor to overheat.
Perhaps if you were to use skype you would have lees issues during calls, as you do not need to share video.
Skype is available on all devices that FaceTime is available for with the added benefit that you can call non Mac and iOS users including Windows 7 PC, Windows Mobile, Android and the link. See it as a more compatible solution.
I discovered a simple hack. If you use Google Hangouts, while in hangouts you can turn the camera off (standard feature). This switches off the camera globally. Even after you've closed the Hangouts browser, next time you go into FaceTime, the camera remains switched off with a message "No camera available. To use FaceTime turn on the camera."