What can I do about this gap between a wooden board and the sink?
I have to say, this is a very uncommon situation. I have never seen a radiator behind a sink or heard of a movable sink table.  I guess you have your reasons, but would it be easier to put doors on the front of the "sink table" so you can access the plumbing etc without moving it?

With that said, the first thing I'd recommend would be replacing the wood board with a rot proof composite or PVC board. I'd add a 2 or 3 inch vertical splash board around the opening by the heater so water and objects won't fall down behind the sink. With the sink in place, maybe you could use a thin bead of acrylic door/window caulk to seal it to the wall and board.  If you find that you must move the sink on occasion, the acrylic caulk is easy to remove and simple to reapply. Another alternative you might try is  a strip of self adhesive rubber pillow style weather stripping to seal the gap between the sink and wall.