Wipe the dirty-looking areas that may be mold with 1 cup of chlorine bleach mixed into 1 gallon of water. Wipe the entire surface with a clean sponge dipped in the chlorine and water. Be sure to wear protective gloves. If the surface area does not wipe clean and the dirty-looking spots remain, then you must assume that the mold growth has penetrated the paint. Go on to Step 2. If the dirty-looking areas wipe off, then you have caught the mold in time and no further action, other than keeping the surface clean and dry, is required.
Remove the old paint from the wood furniture using your paint scraper and heat gun. Remember to use your safety equipment. Pay special attention to removing all the paint from the areas where the mold is growing.
Use sandpaper to sand any rough surface areas and moldy looking areas.
Clean the wood surface again with your sponge and chlorine bleach solution to kill any mold that has penetrated the wood but escaped your scraper. Allow the wood furniture to dry thoroughly for several hours in the sun before applying paint.
Paint the wood furniture with a mildew-resistant paint that also contains a fungicide. This will destroy mold growth, if you missed any, and prevent new growth. Paint wood furniture in the shade, not in direct sunlight, for better coverage.