The problem is that the stove top on Pesach can make your pots chametz much more easily than it could make your pots non-kosher during the year. As a result, even splash back or resting the pot on the stove top could cause problems. Also, the splashback from the stove top could cause your grates (on which the pots rest) to become chometz. This then affects the pots themselves.
Note that the communities that I refer to tend to be Ashkenaz.
The Star-K Pesach Kitchen - The cooktop
The rest of the range (not glass top) should be cleaned and covered
with a double layer of heavy duty aluminum foil, which should remain
on the range throughout Pesach. Please note: Extreme caution should be
taken not to cover over the vent, as well, so as to allow the oven
heat to escape. The drip pans should be thoroughly cleaned and need
not be kashered. The burners do not need kashering or covering, but
should be cleaned.
top of the stove
The top of the stove, however, is a different story. In reality, that
surface is always treif, because it gets splashings of milk and
splashings of meat. During the year, this is not a problem. Why?
Because first of all, you're careful that those splashings don't touch
what you're cooking. And if it does, then it all depends on whether
the splashings are "edible food," and what the proportion is, etc.
Furthermore, if your stovetop is cleaned, then the only problem is
that it has absorbed milk and meat "taste" -- which we assume happened
more than 24 hours ago. So during the year, it's very unlikely that
these splashings can render your food "non-kosher."
On Passover, however, the 24-hour leniency doesn't apply. Something
that was used for chametz a year ago is still chametz. Consequently,
Passover food cannot touch your stovetop and it must be covered. The
easiest way is to take aluminum foil and cover the whole stove top.
Use the heavy-duty kind so it doesn't tear.