Clean the surface of the veneer using a damp cloth containing a mild cleanser diluted with warm water according to manufacturer's instructions to remove any dirt or fingerprints from the furniture. Wipe with the grain while cleaning, and then wipe the surface dry with a clean lint-free cloth.
Fill small shallow scratches that aren't deeper than the top grained layer of the laminate with furniture paste wax. Place a quarter-sized circle of the wax onto a piece of cloth and rub the wax into the scratches in the laminate. Fill the scratches with the wax to the surface level of the veneer.
Buff the laminate surface with a buffing cloth using small circles that concentrate primarily over the waxed area. Blend the wax into the laminate to conceal the scratches, applying more wax if needed to fill the scratched completely.
Fill deeper scratches in the furniture by packing them with laminate repair filler. Use a filler matched to the color of the laminate, mixing several filler colors if necessary to match the furniture. Mix the filler by adding a lighter color filler to a darker one, using a wooden stirring stick to combine them. Pack the scratch with a putty knife, pressing firmly with the flat of the knife to fill the scratch until level with the furniture surface. Wait 24 hours for the filler to harden.
Buff the filler patches with steel wool using small circles to smooth and level the filler with the furniture surface.
Spray a light layer of non-wax furniture polish over the furniture and wipe the surface with a clean cloth to help blend the patches in with the rest of the laminate.