Most white marks or rings on furniture are usually left by water or moisture penetrating through the protective finish (varnish, shellac, lacquer etc.) and then getting trapped below the finish. When this happens the finish in that area appears white and looses its transparency.Details at his link. Also check out the repair tips at RJ Fine Woodworking (mayo, among them) and The Woodworks, Inc.
This usually appears in the shape of a ring and happens on a surface like a tabletop. The ring shaped mark happens because it is usually left by a wet drinking glass or hot cup under which moisture condenses (builds up) on the protective finish that has been applied to the piece of furniture. This usually happens on older pieces where the finish has started to crack or craze. The moisture will seep through the small cracks in the finish.
There are a few methods you can use to remove water marks, and I will list them one by one.
Maybe this advice would work for you.
By the way, condensation is one possibility, but many ceramic vases actually slowly leak through the bottom - you can coat any bare (unglazed) areas on the bottom with epoxy glue to stop that. If the vase is glass, probably not an issue!
posted by Rumple at 12:45 PM on May 20, 2006