In our home - constructed in 1920 - the builders attached the plaster directly to a wood burning chimney that also has the furnace vented through it. (We have never used the chimney due to the flue being stuck in the closed position.) We are thinking the chimney's brick is causing a moisture and mold problem on that wall. How would we be able insulate that area and hang the drywall to keep the moisture from returning and causing this issue all over again?
Kari, this is a very good question and the correct answer depends on who you ask. My guess is that the moisture and resulting mold is being caused by air leakage in the chimneys mortar. My home was built about the same time and has a very similar situation with the plaster on an interior wall being right up against the masonry chimney. I was getting a little bit of moisture on the wall, but my main problem became smoke leakage. I discovered that there wasn't a metal flue in the chimney and the kerosene stove that discharged into it came very close to causing a fire because of mortar that had loosened over time.
Since you don't use your fireplace and there's a flue for the furnace, it wouldn't appear that you have a fire hazard. When you write that the furnace is vented into the chimney, I'm assuming that means there is a metal flue. If that's not what you meant, then your situation could be as potentially dangerous as mine. If there is no metal flue inside the chimney, I strongly suggest having a qualified HVAC contractor inspect the situation as soon as possible.
Even if you have a metal flue, heat could be escaping if there is loose or missing mortar on your masonry chimney. When this happens, the result can be moisture on the adjacent building materials. It sounds like this is what could be happening on your interior wall.
As far as the insulation, chimneys normally require clearance around them of anything that might be considered combustible. The spacing is usually 2 inches, but check with your local building department to see what the requirement is in your jurisdiction. Fiberglass batt insulation without paper backing is not considered noncombustible, but still may be permitted against the masonry chimney by your local building inspection department.
If they allow the batt insulation, the chimney should be air-sealed prior to installation. This should reduce the chance of a fire occurring and ought to eliminate the moisture problem as well. The air-sealing should be done by an insulation contractor to ensure it's done correctly and safely.
Even if you're not permitted to install the insulation, pointing up the chimney's mortar as needed should help stop the moisture. While this project can be done by a homeowner with DIY experience, it might be a good idea to hire a mason as the job requires some expertise. Now that part of the chimney is exposed, it might be the perfect time. I realize that a major chimney repair might not be in your budget right now, but if there isn't a flue for the furnace, that is something that should be addressed as soon as possible. At the very least, get it inspected to ensure it's safe to operate your furnace.