I am assuming that this is a typical bathroom job with no special problems..........such as rotten floor joist and subflooring due to leaks, etc. So, to answer your questions, yes, you could put down uderlayment right over the existing flooring. In fact, if it is asbestos, which if it is sheet goods in not likely, but that is beside the point, you would be required to put the underlayment "over" the existing flooring. If you have a large area of tile (and I am assuming you are talking about quarry or ceramic tile or something on that order) you should make sure there is no "bounce" in the floor. If do you have any "bounce" in the floor, I would reconsider the whole tile thing and go with another flooring method.
If you have selected a thick tile, such as Mexican tile, remember to make dimensional allowances for all the cabinets, doors, fixtures, etc.