Being careful not to fall in your bathtub is important, but don't destroy your tub over it.
By not removing the bathmat after each use will lead to bacteria build up on the floor, etching of the porcelein or enamel and possible rusting of the tub.
With refinished bathtubs you want to avoid bathmat with suction cups. Those little suction cups will pull at the finish and eventually they will cause problems with your newly refinished bathtubs.
One option available when having refinishing done is to have a slip resistant surface installed on the floor of the tub. This is accoplished by adding a ground polymer to the coating and rolling it onto the floor of the bathtub. One the tub refinishing is completed non slip floor is barely visisble.
Refinished Bath Solution makes a bathmat without suction cups. This product is made from high quality natural rubber. It has reverse suction cups to help prevent damage to the refinshed bathtubs. Their mats are elderly, infant and child safe and is backed by a lifetime warranty.
They can be ordered on thier website.