Paint is just one of the many substances that can heavily stain a marble mantlepiece. Marble is a porous natural stone that will absorb stains quickly. The stone is not resistant to acidic cleansers, which will etch the marble surface permanently. It is important to select the correct chemicals to remove paint. Oil based paints may further stain the marble, and you will need to chemically remove those stains, as well. Latex paint will not stain the marble.
Put on protective goggles and gloves.
Saturate a clean cloth with an alkaline paint thinner and apply the thinner to the paint according to the package directions. Allow the paint thinner to sit on the stain for the directed amount of time.
Scrape up the softened paint with a razor blade. Hold the blade at a sharp angle so it is almost flush with the mantlepiece surface and gently press the edge underneath the paint layer until the paint comes up. Do not press the razor into the marble itself, or you could scratch it.
Continue the application of the thinner and the scraping of the paint until all the dried paint is gone. Examine the surface of the mantlepiece to see if the paint soaked into the marble and stained it. This is likely with oil-based paints.
Soak a clean cloth in household ammonia and gently scrub the stain until it is gone. If the ammonia is ineffective, get a new cloth and soak it with white spirit, which is a stronger cleanser. Scrub the stain with white spirit until it is gone.
Do not use these chemicals on a mantlepiece if you have recently lit a fire. Put the fire in the fireplace out, and wait for 48 hours for the stone to cool completely.