If a piece of paper left timber furniture gets moist, such as if a beverage spills on it, that paper may stick to the wood if it is left to dry in place, almost like glued down. The exact same thing can happen if using paper for a coaster for a warm pan or drink. To remove the paper after you have lifted away all you can, rub olive oil within the area and let it sit. The oil will loosen the paper.
Kitchen Quick Fix
To use olive oil as a dried-on paper remover, soak a corner of a paper towel in olive oil, then spread the oil generously over the affected area. The paper stuck to the furniture should absorb the oil within half an hour or so; after that, wipe the loosened paper and the oil away with another paper towel. Petroleum jelly may be used in place of olive oil.