White scratches/spots on wood furniture- how to fix?
I can't identify what the white marks are, but I can suggest some ways of covering it up.

If you have a can of brown shoe polish, you can rub some of that onto the wood and it will help cover up the white marks. Personally, I don't like doing this because shoe polish is wax based and can leave residue that won't absorb stain if you want to refinish the wood later. 

The next easiest option is to buy a bottle of Old English Scratch Cover For Dark Woods. I use it at home and it works well for stuff like this. My biggest complaint about this product is that it doesn't last more than a year. It seems to fade over time and you have to reapply it.

Another option short of refinishing the wood is to buy a liquid stain that matches from the hardware store. You can apply it to the marks and then immediately wipe it off. You don't have to apply a wood sealer on top, but if you do it will last longer.

If you find that the white marks don't absorb anything and seem to be on top of the wood instead of inside of the finish, then you can try removing them with a scouring pad or some fine 360-600 grit sandpaper and retry one of the steps above again.

If this doesn't work, then you might be looking at refinishing the wood. This would mean sanding the top layer of wood off, staining, and then sealing with a polyurethane/polycrylic (read the stain's instructions for which sealer it recommends.)