If it's drywall "mud" (i.e., drywall compound), it is water-soluble. After removing the junk from your tub, fill it with water and let the water sit. Unfortunately, the "mud" will not magically dissolve, but it will loosen enough for you to use your fingernails or a soft scraper to scrape it away. It will be a slow process but it will not affect the tub's surface.
If it is tile grout or mortar, that's a different story. As you probably know, tile grout is very abrasive. American Standard Corp. recommends using a wooden popsicle stick or tongue depressor to remove tile grout. I would add that you want to remove it without scraping the material across the surface. When all is done, American Standard recommends granular Spic and Span® mixed with water to clean the remainder.
Finally, if you do end up with any dulled areas, they can be restored by rubbing with a white automotive-type polishing compound and waxing with a liquid wax.