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This is a messy, gritty job so if you thought you could remove the tub and not get into some major work with the walls and floor area near the tub, think again. However, bathroom renovations are one of the most cost effective from the aspect of your home's resale value so this project will be well worth your time and care. You can remove the tub with minimal damage to the walls, but there will be some work to do after your little demolition project is completed!
Back to your question… yes, you can use a saw for this purpose though it is not the only way to demolish the tub (read on). It's called a cut-off saw and has a very large diameter fibrous cutting blade to make cutting through the cast iron easier. Unless you plan on doing this job often, I suggest you rent one instead of purchasing it. Describe your cutting job to the salesperson to get the correct tool with the correct blade. Don't try to do this with a circular saw with a fibrous cutting blade… the job will take longer, many corners may be unreachable and you may burn out the saw's motor.
First things first… you must disconnect the old drain from the tub. This will probably require you to cut into the ceiling below unless you are over an unfinished area such as a crawlspace or basement. Then you should remove any caulking that is holding the tub to the tiles or tileboard so it is not bonded to the walls. If you have ceramic tile it will be necessary to remove at least one row of tiles to install the new tub so you might as well do that now. If you have a tub surround or a type of tileboard, it can't be saved and will have to be cut to accommodate the new tub.
By the way, I assume your tub is cast iron. If it is a fiberglass tub you can cut it with virtually any saw blade. The easiest and most versatile saw for this type of cutting is the reciprocating saw or "sawzall". Blades are available up to nearly a foot in length, making it easy to cut most anything. Just don't cut through the floor or your plumbing!
Start with two cuts across the width of the tub, dividing the tub into thirds. Once the cuts are complete, you can remove the center piece. If it is still bound, it can be broken into pieces with a sledge hammer. Ditto with the end pieces. Don't go crazy with the sledge… just do what is necessary to remove the tub! IMPORTANT: Cover the tub with a heavy tarp to keep pieces from flying around. USE EYE PROTECTION, TOO!
Early on I mentioned that there might be another, easier way to remove the tub. And maybe a little "stress relieving", too! Well, the fact is that many if not most remodelers would not use the "finesse" I describe above. They will simply cover the tub with a tarp to keep pieces from flying and just whack the heck out of the thing with a sledgehammer. If the plumbing breaks, it isn't a really big deal since the project will require lots of modifications to the water supply and drains anyway! So pick your poison!
Because new and old tub dimensions may be different, you will have to do at least some wall work to install the new tub. If it is the same size as the old one, it may extend to the studs, not the finished walls, making it difficult to get in place without some persuading. If the new tub is smaller, fitting it will be a little easier. To fill the resulting gap, you can either 1) make a shelf to fill the gap (on the non-plumbing side) or 2) erect a floor-to-ceiling wall on either end of the tub. The floor-to-ceiling wall is really the better choice, especially if you also use the tub as a shower. The shelf will tend to collect water and be a leak waiting to happen!