Do you have to pull up vinyl flooring before tiling bathroom floor?
There's usually some type of underlayment beneath the vinyl.  When I did my kitchen floor, it had luan board.  I would remove the vinyl.  You really just need to remove the luan board and the vinyl comes for the ride.  Then you can put down cement backerboard or even the rubber membrane.  

Unfortunately/fortunately, the luan board is stapled down, which is easier than if it was glued down, but there's a lot of staples.