Dampen a clean cloth with fresh lemon juice and rub it into the burn mark. For minor or recent burn marks, the bleach-like qualities of the lemon juice may be enough to remove the mark.
Combine 1 part baking soda with 3 parts water and stir the mixture to create a paste. Use a toothbrush to spread the mixture over the burn mark and rub it in. The paste itself should bleach the area and the abrasive bristles of the toothbrush should help lift the mark.
Rinse the sink well to remove the baking soda paste, then gently work at the burned area with a piece of fine-grain sandpaper. This method works best on natural stone and composites but may leave a depression in the sink which you will need to fill.
Clean the sanded area to remove any dust or dirt, swabbing it with a cotton ball soaked in rubbing alcohol.
Fill the depression with all-purpose epoxy filler or find a bathtub repair kit that matches the color of your sink.
Allow the filler to set for at least 10 minutes, then gently sand the surface with fine-grain sandpaper until it is completely smooth and level with the rest of the sink surface.
Apply the paint to the filled area. If the color of paint from the repair kit matches your sink closely enough, the repair should not be noticeable. If you cannot find a color that matches completely, however, select a paint slightly lighter than the sink because a darker color will be more noticeable.
Allow the paint to dry for a full 24 hours before using the sink.