Firstly, you should dry clean your antiques regularly with a dry cloth for dust.  If there is dirt that requires attention, use warm water on a cloth (not sopping wet) and clean the surface.

If you have a piece that requi9res more cleaning, consider using a mild cleaner, such as dish soap and warm water.  Always test your cleaning solution first on a spot which will not be noticed.  Such as the back of a leg or underside of your piece. The cleaner should not cause any white haze or dulls the finish.  If so, you can use it further. 

You want to ensure that your piece is fully dry.  I always permit my pieces the time to dry overnight at room temperature.  If your piece was soaked fully, you may want to even wait longer.  Your piece should be fully dry to the touch and the wood should not feel soft from water evaporating.

I have been successful with removing stickers and plastic tack paper from shelves and drawers with my heat gun.  Set your heat gun to a low temperature and heat the surface of the sticker or tack paper.  As the glue loosens you can slide your putty knife between the substrate and pull upwards slowly.

The glue will remain after you have removed the sticker or tack paper.  This can be removed with white spirits and steel wool. Or Rubbing Alcohol. It’s messy and stinky work.

I hear that Goo-be-Gone would work well. I have not tested this, as it is not available in my area.