What kind of drill bit would I use to grind down the end of a protruding screw?
A protruding screw head is made of steel, so it will be moderately hard. If the screw is brass, it would be easy to grind down. You have several options.


With 1/4 inch remaining above the surface, I might first cut it down rather than grinding it all down. A piece of hacksaw blade will suffice, held in my hand flat against the surface to avoid damaging the wood. Leave a little stub that can be dealt with easily.
A grindstone bit, for use in an electric drill. This can be found in any hardware store, probably made of aluminum oxide, and possibly a 1/4 inch shank. Smaller ones for a Dremel tool will have a 1/8 inch shank, but I'd get the larger one. It will last longer, and fit in any electric drill, so no Dremel required.
You can use a high speed steel or carbide burr to do the grinding too. But aluminum oxide is harder than steel, and far cheaper than carbide.
Use a bimetal blade in an oscillating multitool. While I have found many uses for this tool, the blades are incredibly expensive and will wear out quickly when cutting steel. (There are titanium coated bimetal blades that will last longer, but they are not at all cheap, and may not fit on every model multitool. For example, no titanium blades are available for my Porter Cable tool.)
Sandpaper on a sanding block, as suggested by one answer is an option, but sanding down a 1/4 inch steel stub will take some time and probably tear the paper into shreds (as well as the sanding block.) So I would definitely shorten the stub using a hacksaw blade first. A belt sander will also do the job here, again be careful as it might tear the belt.