Do: Use trivets and hot pads under pots, pans and dinnerware.
Yes, you can take a hot pot off the stove and put it right on granite, quartz or soapstone without any problems. Not true for all other surfaces. But you must consider other issues as well.
Grit that gets trapped between the pot and the countertop surface may scratch the surface–even granite. Granite is very hard and can take tons of abuse without any damage, but it does contain some softer minerals that could be chipped or scratched. The likelihood is pretty slim, but it is possible. And ALL other surfaces are softer that granite. Better safe than sorry.
If it does happen, don’t fret too much. Most chips and scratches can be easily repaired, but it’s best to avoid them by following the granite counter top care tips.
Also, once you remove the hot pan from the countertop the surface will be very hot and may burn.