first are we talking about vinyl tiling? not ceramic correct?
Then, if the old tiles are not damaged and loose or broken, then you can tile directly over them. use a floor levelor or float to smooth out any areas where tiles are missing or sometimes you can remove that entire tile and place down another good one to fill the gap for the new flooring.
Then, you can also remove the old vinyl tile. This will work also, but you do have that sub floor that could have glue and buildup that has to be removed so that the new flooring will lay flat and smooth.
Remember, any slight, slight imperfections in the floor or subfloor will mirrow exponetientially on the new floor and if its a rise, that area will wear very fast.
If you go over the old tile then make sure that the pattern of the new tile is not congruent to the old, make the seams overlap and not identical.