I recently had old tile backsplash removed down to the studs. Is it ok to use drywall instead of cement board on a kitchen backsplash behind the cook top, there is no running water in the area, the drywall will dead end into the countertop and be covered with tile.
Is it "OK" - yes, plenty of tile is installed on drywall (assuming you are planning to retile? If not, well, you don't even need a "backsplash as such.") Plenty of kitchens with painted drywall there that do just fine.
Is it a good idea - I would say no, because backer-board is a superior base for tile .vs. drywall, and tile is an expensive material involving expensive labor (if DIY, you can count that part as time rather than money) as well, so saving a few bucks (a tiny fraction of the cost of the tile, thinset or mastic, and grout) by using an inferior substrate for it is a poor choice in the overall project cost / longevity tradeoff.
But sure, it's "OK" according to any number of houses built by people who don't plan to live in one of them, just have them hold together until a not terribly long warranty expires. Various tile associations approve methods of doing so. It's not considered unacceptable to them.
Do follow a recommended procedure for tiling over drywall, if choosing that route. One thing I glanced at mentioned not using drywall compound on drywall intended to be tiled, for instance, as it may reduce the tile bond to the drywall.