It appears to be a universal Ashkenazi custom to cover the area of the stove top between the burners, underneath the grates with aluminum foil. (The grates and burners and cleaned and kashered.)
As your Pesach pots or food served do not come in contact with the underlying porcelain surface, what difference does it make if it's covered? i.e. If you would never take food that splattered out of the pot, and eat it or return it to the pot, then is covering the area with foil really needed?
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.
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.
