Most of the time, it's no real trouble removing soot from a brick fireplace. All you need is chemical cleanser and a coarse brush. However, when the brick fireplace is painted, you must use great care or you'll likely damage the finish, or at bare minimum, cause some discoloration. More often than not, people get too aggressive when trying to remove soot stains from a painted fireplace. This often results in the entire fireplace having to be repainted. If you need to remove soot from a white painted brick fireplace, there are a few techniques you can use that may save you from having to re-coat the entire surface with a fresh coat of primer and paint.
Press the children's molding dough or Silly Putty against the soot stain, then pull the dough away. It should remove some of the soot. Repeat until all of the soot has been removed. If this fails, use dish soap and warm water.
Mix ordinary dish soap with warm water. Use a coarse scrub brush to apply the soapy water to the soot stain. Scrub in a circular counter-clockwise motion. Rinse the surface with a wet rag. Proceed to Step 3 if any soot remains on the painted brick fireplace.
Mix the white vinegar with water (1/3 vinegar, 2/3 water). Use a coarse scrub brush to apply the vinegar to the soot stain. Scrub in a circular counter-clockwise motion. Rinse the surface with a wet rag. Repeat until all the soot has been removed.