Can moisture from a chimney cause my furnace to shut down?

Since we had the Noreaster three weeks ago, it seems now that everytime it rains out -even drizzle and fog, the furnace shuts down. I had Sears come into the house and I was told that the furnace is fine and the thermostat is new and working ok. The furnace has a sensor switch that when moisture hits the furnace the switch will power the furnace down. Sears told me to have my chimney checked out. They said that a sweep is in order but I am wondering could my chimney have a leak somewhere that the entire chimney might have to be replaced? The house is 102 years old and inside the chimney is slowly losing its pieces up in the attic. But I can't see how the exterior would allow mositure in it. I have the top of the chimney covered to prevent animals from going in. I have one of those mesh nets up there. What could it be and how much will it cost to repair my chimney?