Can a problem with heater stack liner cause outside chimney mortar to crack and fall out
Rain a day after pointing work will help, not harm, the cure. Freezing weather would harm the cure.

High efficiency furnaces output low-temperature flue gas - more heat in your house, less up the chimney = higher efficiency.

Flue gas is always moisture-laden, and starting at a low temperature and then dumping into a cold masonry chimney means the moisture will condense out inside the chimney, rather than exiting as a gas and condensing outside the chimney. Depending on the gas quality, this condensate may be somewhat acidic, thus corrosive (attacks mortar, steel etc.)

It would be worth checking your furnace manual for chimney recommendations - many of the very high efficiency units actually want to be vented into PVC that's sloped to let the moisture drain out (or at least has a drain at the bottom, if vertical.) The temperature (on those furnaces) is so low that PVC is perfectly adequate for the temperature, and it has good chemical resistance. A stainless steel liner would be another possible option - a masonry liner might well not. In any case, what you use should comply with what the new furnace wants, which may not be what the old furnace had (or needed.)