Hans Wegner RY 20 Clean up  

That is very likely a water stain. Oxalic acid is the perfect product for that. I can buy it at my sherwin Williams paint store in a plastic container labeled as "wood bleach." It is also one of the ingredients in Barkeeper's Friend that you can get in the cleaning aisle anywhere (if you use this, do NOT scrub it).
I like to use Murphy's oil soap to remove an oil finish. Use it as hot as you can stand, and scrub the the grain. You could also use steel wool to aid your scrubbing. Or you could sand, but this piece is veneer, so do NOT sand throughout the veneer.
Then, when the oil has been removed, mix up some oxalic acid, brush it on the area, I like to feather it out a bit, or brush it on to an entire surface. By the time it is dry, the stain will probably be gone like magic. It reverses a specific chemical reaction that causes water stains (iron and tannins in the wood get brought together via water and make a black substance,which is the water stain), and it is likely there is some low degree of that across the entire surface, which is why I do the entire top, or feather it out. Otherwise you can end up with a lighter area.

I've had mixed results with Barkeeper's Friend. On some pieces with very dark black stains with well-defined margins, it completely removed the stain with no bleached look at all in the end. It was as if the stain never existed. And that's with treating only the stain, not feathering the solution onto the unstained areas.
Other times, it bleached the wood a lot. Still other times, it didn't lift the stain entirely. Sometimes it works really quickly, other times it takes a long time. I guess it depends on the stain, the wood, and the durability of the finish.
I have also just tried rubbing it in with my fingertip and wiping it off after a few minutes to check on the progress, then repeating as needed.
I used to use Murphy's Oil Soap to remove old oil finishes like Leif describes above but found it dried the wood a lot and would sometimes raise the grain. Now I just use Star-Brite brand teak oil (Ace Hardware carries it) and #0000 steel wool. This oil seems to have less solvent in it and no varnish, and it just softens and lifts the old oil so that you can wipe it off. For really filthy wood, like where it's black with grime, Murphy's is good but I use it straight, not diluted. Wipe it off completely and then rinse thoroughy with wet rags and dry immediately. This is still drying to the wood but it's quicker than working on it with teak oil. (Make sure the soap is completely rinsed off. I once did a half-assed job of rinsing and then the oil finish reacted with the soap and got all cloudy. Ick.)