Damp patch on the chimney wall
...This is one school of thought: seal the chimney to prevent air flow. By preventing air flow, you also prevent the flow of moisture the air carries with it, and the formation of condensation on the colder, interior face of the flue. FYI, the insulation you push up into the flue works simply because it blocks the flow of air; insulation 'warms' nothing. What it does is slow heat transfer from warmer to cooler areas.

The second school of thought is to increase air flow. The theory is that, with constant air flow, moisture that settles on the inner surfaces of the flue will be evaporated.

For this system to work, the flue has to pull in dryer, ambient air from a lower floor; this requires creating an opening below the damp area. This is the preferred method in Britain, for example.

Incidently, houses built in the US Northeast throughout the 1800s had just such an air circulation system built into their exterior walls. In old houses, you have a cavity between the brick and the plaster walls and that cavity extends from the basement to the attic. Air in the attic is heated and exits, creating a partial vacuum which draws air in the cavity upwards. As the air is drawn upwards through the wall, it is heated, which increases its capacity to absorb moisture. This wicking system removes moisture from the inside surface of the brickwork and prevents moisture setting into the lathe.