I posted a message with this same subject line back in September. I wanted to let everyone know how it worked out.
The problem, to recap, was that 30 years ago, my father-in-law had a heirloom family table professionally refinished. It came out great, and to protect it, my mother-in-law covered it with a tablecloth. But unfortunately, the tablecloth she chose had a foam rubber backing, which stuck to the new finish. For all the years since, the top was disfigured by a layer of black gunk, and everyone was just sick about it.
My wife and I now own the table, and I asked the group how to remove the black layer. My first thought was to use a heat gun, but everyone wisely talked me out of that. I did as suggested, starting with the mildest possible solvents, and working my way up. I was getting discouraged when mineral spirits didn't work, but lacquer thinner did the job!
It was in pretty nice shape underneath, but you could see some "patterns" of where the old foam had been. I decided to go the extra mile and sand the top. I was able to get a stain that matched identically, and gave it two coats of varnish.
It looks fabulous. It's quarter-sawn oak -- who'd have known?
But here's the best part of the story. My father-in-law is still living. My wife invited him and his two sisters to our house to celebrate the birthday of one of the sisters. (These folks are all in their 80's now.) When my father-in-law looked into the dining room, he gasped ... and tears came to his eyes. He and his sisters then got to have a birthday party around the same table where they sat when they were children.
Who says woodworking isn't a great hobby?
Thanks for the great advice, everyone!