The problem is that most finishes have a plasticizing agent in their chemical composition to keep the finish from becoming brittle and shattering when dry. When the plastic cover was used on the table there was a chemical reaction and it actually broke down the finish. This can create either the weird oily looking surface that you see here, or it can actually melt the surface so that you can pick the finish off with our finger nail.
Usually the only way to fix plasticizer damage is to completely strip and refinish the table. But before we took that step we tried a less extensive procedure, just in case. In this situation it was enough. We began by removing the top layers of the table's finish where the damage was. We then stabilized the finish and top coated it with a new water base topcoat in a 'Matte' sheen. Our client is very happy with the results and has also learned a little lesson about plastic and chemistry.