Remove as much of the fabric as you can with scissors so that the only fabric that remains covers the sticky substance that is affixing the fabric to the table.
Mix 1/2 cup white vinegar with 1/2 gallon warm water. Dip a rag into the solution and then, with your forefinger, gently rub the wood in a circular motion until the sticky substance dissolves and lifts from the surface. Be careful not to saturate the wood, as excessive liquid can damage wood. It’s better to make repeated attempts with a damp cloth than one attempt with a saturated one.
Mix 1/4 cup gum turpentine and 3/4 cup boiled linseed oil in a bucket. Dip a rag into the solution and gently rub the wood in a circular motion until the sticky substance dissolves and lifts from the surface. Again, you may have to make several attempts, depending on the durability of the sticky substance.
Spray the oil-based cleaner-solvent onto a rag. Rub the rag into the wood in a circular motion until the sticky substance dissolves and lifts from the surface. You can find it in spray, spray gel and aerosol forms.
Clean any sticky residue left on the surface with oil soap, a gentle wood cleaner. Mix 1/8 cup of the soap with 1/2 gallon of water, or follow the directions on the bottle, and apply the mixture sparingly to the table with a rag.
Finish the job by polishing the table with your favorite furniture polish, or make your own lemon polish by combining 1 quart mineral oil with 1 tablespoon lemon extract, applying it to the table and then buffing it to a bright shine.