Should a roof drip edge be placed above or below roofing felt?
It depends on where the drip edge is.  At the side, or gabels, of a roof, the drip edge goes over the roofing felt.  At the bottom of the roof, the drip edge goes under the roofing felt.  I have no idea why Ask This Old House did the drip edge the way they did.  The main purpose of underlayment is to keep water away from the decking if it gets under the shingles.  Water will pass right under that drip edge.
But don't take my word for it.  Your shingles will come with instructions that say how to place the drip edges.
Example Installation Instructions
Timberline® Shingles Installation Instructions (PDF)



Oakridge® Shingles Installation Instructions (PDF)



Example installation instructions located using a simple google search. Products listed are not recommended nor endorsed by the author of this post.