Thanks for the contributions, everybody!
I said I'd report back on my Olive Oil experiment. There's been a drop on a saucer in the kitchen for almost 3 days now, and it's in just the same condition as it was when it came out of the bottle. So it dries very slowly, and as Bugbear indicated above, probably doesn't dry out completely but does go stickier over a period of weeks or months. We also know that sometime between weeks and about 2000 years, it goes rancid (thanks Harbo and RichardT).
Overall conclusions - it could be used on non-porous stones such as Arkansas and slate, but it might be wise to wipe it off fully after use. Putting it on porous man-made stones might not be such a good idea, as it'll soak in, partially dry and clog the stone over a period of weeks or months. Better options are available for use on porous stones, such as light mineral oils (baby oil, light lubricating oils, and oil/paraffin mixtures, for example).