Same problem. I have the phone for 14 months now and it suddenly makes this fan noise and the screeching sound.
From the beginning the video was also very grainy and noisy with blue pixels al over the place in unnatural lighting.
Going to send it back under warranty, however, already considering asking for money back. As I've just read in a few forums, most of the times Microsoft Nokia tends to be horrible in customer service. And the problem doesn't get solved.
This forum doesn't allow me to give you the link so google this:
Lumia/Problem-i-have-nokia-lumia-1020-Problem-is-sound-when-recording/
/watch?v=3NdYz8o8EKQ
/watch?v=ClbhdgJsAKE
/watch?v=0_A9YP_MW4Q
nokia-lumia-1020/256979-2.htm