Jerome I do not think you will need to remove that tub of yours. Most likely it is sitting pretty and once you start removing tiles you will need to support the tub with little 3/4" or 5/8" shims. Most plumbers will leave the tub off the tub deck this amount so the tiler can slide his/her tile under the tub.
As for new handles and such that is a question for the plumbers. Adding new handles to an old filler is normally not a hard process as many times the parts are universal but on occassion you might need a reducer or transition piece. Your plumber can tell you. Get the tile busted off and open things up.
If your tub was installed with cement and no bond breaker (we use poly here in Vancouver) it is installed for good and removing it might damage the tub.
I would waterproof your tub deck if you have not already planned to do this. Depending on how the tub deck was built this can be a easy or hard task. Get it opened up more and repost some pictures for use.