Not easy to provide reasonable answers without knowing a few more things, such as how wide the door opening is, how tall the tub is (that will be the narrowest dimension), and what the tub is made of. Most tubs are less than 16" tall, which is an easy fit through any interior doorway. Standing a tub on its strong end is usually the best way to get it into a small bathroom, resting and sliding it on a carpet scrap to prevent chewing up the floor. Plan on having some help with the new tub, as carefully dropping one into a tight space can be tricky, if not impossible, for one person. Especially if the new model is cast iron, some of which weigh more than 300 lbs.
In a previous house, I installed a large acrylic whirlpool tub without any help. It was a 2-person model, about 4' wide and slightly more than 2' tall. Didn't have to remove any wall or doorway framing material, either; just pulled all hinge pins and temporarily removed the interior doors that were a tight fit.