Tap on the tub. Does it sound like it is solid or hollow? An acrylic tub is light and will sound hollow inside, like you are knocking on a door. A stainless steel tub will make a ticking or clacking sound. A cast iron tub will sound like a deep gong, just as if you hit a cast iron frying pan.
Look at the bottom of the tub if possible. A cast iron tub will be black, and a steel tub will be metallic, although it may be covered with a layer of paint. An acrylic tub will look like coated plastic.
Try to lift the tub. If you are removing the tub, it is easy to tell if it is cast iron. A cast iron tub is at least four times heavier than a steel tub.
Ask your neighbors and the people who owned the house before you. If you live in a housing complex where all the houses are similar, there may be records that state what tubs were installed in the complex. If you are just moving into a house, ask the previous owners who installed the tub and when, and ask them what the tub is made of.