When you have your bathtub refinished (also known as reglazed or resurfaced) the new surface will be slippery. Bathmats will ruin your new surface and void your warranty. Here is an example of a tub I had to refinish for a customer. This tub had a very good, professional, refinish job done to it. It was in excellent condition with the exception of the one area you see where the bathmat was.
This customer paid the extra cost to have the tub prepped for a second refinish, and then paid the standard refinish cost all because someone used a bath mat when they could have had the slip resistant floor installed for a minimal cost.