Henry is right on about the single edge razor blade. You steel or cast iron tub will easily endue this process with only some black scuffs if your blade became dull during the job (change blades frequently!). These scuffs clean off quite easily.
If you have a an acrylic tub be real careful with the razor blade or your tub will be trashed! These plastic tubs are basically junk compared to metal tubs so I hope that you are not trying to recaulk one of them. If you are, try cutting straight down the tile wall to the tub surface then gently prying the old caulking off with the razor or a sharpened spatula (a handy tool). Lay the spatula almost flat on the tub to prevent it getting an ngle of attack on the tub - it's the caulk that has to go. Good Luck with your repairs!