Using FaceTime without the video 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?
Yes — if you simply minimize the FaceTime window (⌘ Command+M) or hide FaceTime (⌘ Command+H), video will pause while audio still functions as usual.

(Similarly, on iOS, pressing the home button will pause video and allow you to use other applications while audio still works.)