Do I install the closet flange on top of the subfloor, or lay the new hardwoods, then put the flange on top of the hardwood?

The flange should install on the finished floor. Good luck, tom

Tom is correct. But if you have problem with removing your closet ring - than you can leave it on sub floor and Oatey "closet ring extention" rings to build it up to the new finished floor level.