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I'd try MS first, I don't know of a finish that it will harm but as you said test first. If that doesn't work try alcohol, or Murphy's soap in warm water. If that doesn't work try mayonnaise ( you need to let that sit for awhile). I haven't tried mayo but several folks here swear by it and a friend tried it and it worked well for him.
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It's possible (maybe not likely) that the foam backed table cloth actually had some chemical elasticizers in it that softened the finish. If that happened, I'm not sure cleaning will be possible. Some plastics have them to stay flexible, though it's not as bad now as it used to be.
I started with absolutely nothing. Now, thanks to years of hard work, careful planning, and perseverance, I find I still have most of it left.
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Pedro,
Based on the age I'll bet that the finish is shellac. Heed the advice above to avoid alcohol, likewise for acetone and lacquer thinner.
Mineral spirits (MS) is usually a safe bet. It will not dissolve shellac, lacquer or cured varnish. You could rub gently and carefully with 0000 steel wool or the plastic equivalent dampened with MS.
Mineral oil dissolves whatever mineral spirits will dissolve, and it is not volatile and will not smell up the room. If you get the gunk off with mineral oil you can then wipe off the oil with rags and then a rag dampened with mineral spirits.
If a plasticizer did soften the finish you are probably SOL and will want to strip it and refinish. If you can get down to a hard dry surface and want to partially refinish, dewaxed shellac (Seal Coat) reputedly sticks to almost any substrate.
Good Luck
Doug