Should I remove nailed-down plywood before laying tile?
The plywood is there to make the vinyl the same height as the rest of the flooring in the house. If you add backerboard + tile on top of it, you are going to add somewhere in the range of 5/8" to 3/4" of height. That is enough to be a tripping hazard, and it also wouldn't look very nice. 

Pulling up the plywood and the putting down the backer board is the right thing to do.