You can attempt to sand and polish the damaged surface, but the repair will likely be easily noticed. If it was not too badly discolored, try putting several coats of clear urethane over the repaired spot to build up the height after sanding it down. This will help to hide the repair.
Although a laminate countertop can be repaired if the discolored burned area is just on the surface, it may be difficult to recreate the original surface finish and appearance. You can try sanding the spot to see how deep it goes and then talk with a kitchen remodeling contractor about a possible repair.
Another relatively inexpensive method to repair the damaged laminate is to remove that spot and replace it with ceramic tiles. This can actually look quite nice and decorative and will certainly be unique to your kitchen. Since ceramic tiles are heat-resistant, this will be the perfect location to set hot pots and pans without damaging the countertop.
Measure the damaged countertop area to get an idea of how much ceramic tile you will need. Be sure to make it large enough to fit your largest pot with a couple of utensils around it. Keep in mind it does not have to be square. Let your creative juices flow when selecting the size, shape, and color of the ceramic tiles.
If you have not laid ceramic tile before this, it might be wise to select larger tiles. Although it is not difficult to lay the tile, using larger tiles allows you to space them farther apart without it looking odd. With wider spacing and finished grout lines, slight misalignments are not as apparent as with narrower grout lines.
All ceramic tiles will resist heat and be adequately strong to support pots, but the surface of ceramic tile, even though it is hard, can be scratched. If you are very careful and your children are older, a glossy tile surface is easy to keep clean. If you have younger children who may be less careful about scratching or scrapping the tile surface, choose a dull glazed finish. Scratches on this surface are less apparent.
Locate the tiles in their exact positions on the countertop over the damaged area. Draw a line around them on the countertop with a marking pen. Next, draw another line around the first line adding on the width of the grout line you are planning to use.
Rent an electric router at your local tool rental outlet. Purchase a 1/4-inch carbide router bit with it. This will be used to rout out the damaged laminate surface to create the recess for the ceramic tile. Your goal is to have the finished tile surface even or just slightly above the laminate surface. Since you will need space for tile adhesive under the tiles, rout the depth to the thickness of the tiles plus 1/16 inch.
Test fit the tiles in the routed area. Remove them and coat the recessed area (don’t forget the sides) with urethane sealer. Apply the tile adhesive and place the tiles in place. Once they are secure, finish the gaps between them with grout.