With so little information, this is like a stump the chump from BPI. A picture would be nice...
I would think it is likely that the furnace was over sized for the demand and short cycled. The short cycling will not heat up the chimney enough to purge the condensation so the water is absorbed into the masonry. Repeated cycles of freeze and thaw will cause the mortar and the bricks to fail.
It is oversized only if you have performed a load calc. Otherwise who knows if it is right or wrong sized. Even if it is the correct size, an 80 % furnace vented into a an unlined chimney will condence. That will happen at a greater rate when the chimney is in a cold attic. If the house was sealed, then the attic is colder yet and the condensate will condense at a faster rate.
It is a classic. The chimeny was probably failing before the air sealing and before the furnace was installed. Assuming that there was a furnace there before. The change in the house probably made the falure more rapid. It will happen even if the furnace is correctly sized, under or oversized. The system was just installed incorrectly.
This extremly accurate opinion has been reached by taking a question with no facts provided to its logical conclusion. If you can measure it, you can quantify it, if you can quantify it you can compare it to what is normal and only then can you diagnose the issue and apply the correct remedy.
If you want a factual answer ---- provide facts
Eric, if you provide a little more information, we can sleuth the answer for you.
-- Age of house?
-- what kind of chimney materials (Brick, stone, CMU, etc.)
-- Climate where this house is?
Chances are if it's an older house (70-100 + years), then the mortar was already failing. We see these everywhere in the Bay Area, and the furnace has nothing to do with them. There should be a flue liner, if this chimney connects to other sources, like a wood burning fireplace.
A natural gas furnace vent doesn't put any pressure on the chimney; the water from the combustion wouldn't affect the mortar unless the chimney was blocked and not venting properly -- but then you'd have CO problems, which you haven't indicated.
Pictures and more information would be helpful.
the house was build in Kansas City MO in 1903. 4710 Degrees days with about 65 days gets under 32'. All stone with a 18"X18" clay liner that had a 850,000 BTU boiler now just a 120,000 up flow 80% with that main floor now has a heat loss of just 42,000 BTU at 0' out side. So the furnace is 3 times over sized and the flue should be 5" liner. The stone chimney heat up about one foot each time its start up. The dew point with flue gas is 255' stone chimney does not get up to temp unless some one turns up the heat or leaves the main door open. The chimney stays wet and some time goes though the freese with lack of being in building. After 2 year with a 80% furnace the chimney started to fall. The boiler had a 7" flame to light the burners and to keep the flue hot.