How to clean a stainless steel gas cooktop without harming the surface

Use some Flitz. I used to have a tube of the paste that worked really well on stainless knives. I need to pick up some more from the hardware store.

If the tub material is fiberglass (and most commonly are) the best way to remove any blemish is with an orbital sander and some 300 600 and 1000 grit Wet and Dry sandpaper. Using a spray bottle filled with water wet the stained surface. Starting with the lowest grit sand and keep the area wet. Let the sander do the work. No need for hand pressure. Watch how fast the stain is being removed (it should go quickly. Move through the grits and spray to keep surface continuously wet. The paper is of such a fine grit it will not leave any discernible scratches. When completed apply a cleaner/sealer for fiberglass.