The remote was sitting on a wood shelf for more than a year, at least thirty feet from any food items. The shelf is pine that was stained a dark color. I have probably fifty linear feet of that wood, with that stain, and none of it has ever been sticky in any way since it was stained. In 1970.

The portion of the remote that touched the wood is sticky in a way identical to the parts that were not touching the wood. So it's not the wood, nor its stain.

Thanks for the ideas. I'll bet Bestine does the trick. Now to find less than a gallon of it for sale somewhere....