It is something to worry about, sorry to say.
Yellow Tangs are notorious for getting what is known as "Black Ich." Black Ich isn't caused by a single organism, but any number from a group of flat worms that burrow into the skin of the fish. You can't see the organism on the fish.
The black dots you see are actually a skin reaction called hypermelanisation to the worms' affect.
These worms fall off the fish and complete their cycle in the aquarium, hatching more eggs to find more hosts. Although Yellow Tangs seem to be the most prone, many different kinds of fishes can get it (and carry it into your tank).
This is one of the organisms that can, in small numbers, be gotten rid of by doing a freshwater dip on newly acquired fishes, then holding them in quarantine, so that such worms don't make it into your display tank. It's one of the reasons I recommend a freshwater dip on all newly acquired marine fish, followed by quarantine:
Freshwater Dip for Marine Fishes
Although there are two primary treatments (multiple freshwater dips and multiple formalin dips) I recommend using formalin when you are at the stage you see.
It is best to 'punch' those worms out with a tough treatment. The fish will be fine through this.
Move the fish to a hospital tank. This is important so that the fish will be easy to catch. What you have to do is multiple dips in formalin. For the formalin dip procedure, read and follow this advice:
The longer you leave that fish in the tank, the more worms you'll be letting reproduce. I'd suggest moving quickly on the removal of the fish.