Can I remove the bathtub -- just the tub -- without shutting off the water supply?
The drain parts (the drain assembly is called a waste &amp; overflow unit) can be disconnected from above, there should be no other direct connections to the tub.
Just unscrew and remove the overflow trip-plate and disconnect it from the stopper mechanism (called the bucket &amp; wire assembly). The drain strainer can be removed from the tub drain shoe by unscrewing with a special tool called a spud wrench.
The main problems with this idea are:

the tile/wall treatments (as mention by other commenters) are usually configured such that they overhang the integrated lip of the tub, which means the lowest tile course would have to be removed (see pic below).
some (many) bathtubs are cast iron and extremely heavy, making removal without significant alteration difficult. On more than one occasion I have had to demolish the tub in-situ to get it out.