I am sure that I have done a TV show on this product, but for the life of me I cannot find it and people keep asking. There is a special category of re-finishing product, often called 'furniture re-finisher' (not furniture or paint stripper) but variations are made by several different manufacturers and it is easy to find in the stores. It is basically an acetone based product but works better than straight acetone.
It will do absolutely nothing on a modern polyurethane plastic type finish, but does miracles on any lacquer finish. Lacquer was used on most of the antiques and on some new factory applied finishes.
So if you have a water mark in the finish, or a rough dirty finish where the stain below as well as the wood below is still in good shape, you can use Furniture Re-Finisher. It will not strip off paint, nor will it strip other finishes down to the wood. It has limited use, but is great where you can use it. If you have scratches into the stain or the wood, this will not do the job: look at Eleminating scratches from furniture.
There is no point in buying a can of re-finisher if it won't do the job you are trying to do, so get some acetone first; either some straight acetone or fingernail polish that does not say 'acetone free'. Apply a little with a q-tip and see if it affects the finish. If it acts like water on the top - furniture re-finisher will do you no good on this job. For haze in a plastic finish, look at Wipe-On-Poly. If it tends to melt the finish, then go for it.
Put the Furniture Re-finisher in a small bowl. Take a fistful of very fine steel wool and dip it into the bowl. Now gently 'clean' the furniture with this. Return frequently to the bowl where you will dip into the re-finisher and then wring out the steel wool. You will see all the ages of grime collecting in the bottom of the bowl. For a water mark, you should work on removing the water mark itself and then blend the work out over the entire surface to not have the 'clean' area stand out.
What it does is dissolve the finish. As you work gently with the steel wool, the wool fibers cut and pick-up the old finish. Be careful not to scrub down into the stain or you will have to completely strip and finish the piece of furniture. It also tends to spread a bit of the old lacquer out evenly over the surface. When all is refreshed and dried you should put on a coat of anything you like: lacquer like the original finish, or modern polyurethane.
It is called Furniture Re-Finisher - not Furniture Stripper.