I have a 2nd home, about 80 years old. The walls are plaster style, probably with no insulation installed. The heating is the floor furnace style. Recently I noticed some black mold on the bedroom walls that is on the exterior side. There is no plumbing (water) in that area, so no water leaks. Should I remove all plaster, and install insulation and new sheetrock. Thanks in advance

That would probably be your best bet. While removing the plaster or any kind of demolition work where the mold is, it would be a good idea to wear a respirator to avoid inhaling any airborn mold substances as will happen when the mold is disturbed. If you notice some mold within the frame work when you remove the plaster, using a product such as sporicide could be used to clean and kill the mold to the wood members.

Are you sure it's not just dirt/debris from the furnace vents.

If not just around the vents, is it just a top layer mold,or does it go through the entire wall?

Any windows near this area?

What is the siding of the house?