Do you have an attic above that? Maybe check to see if you have a leak around your chimney.

Perhaps the first step would be to strip off the paint down to a nice clean finish. That might make it easier to see where this might be coming from. Whatever it is there must be moisture coming from somewhere to facilitate the growth.

Thanks! I plan to clean and paint the wall, sealing the plaster and molding as I go.

My concern right now is to get a sense of what is causing the residue in case there are larger issues at play. The proximity to the chimney makes me wonder if it is somehow related to moisture moving up the chimney (the hot water heater vents through there), but I really don't know anything about this!

As I under it, efflorescence is usually white and this crud is brown-yellow. Maybe it's debris or dirty being pushed through the brick or lath-and-plaster by moisture? I've ruled out mold because this is dry and crumbly - unless mold can sometimes be like that also?

You need to figure out where the moisture is coming from. Cleaning, painting etc will be pointless until you identify and repair the source of the problem. Start with the flashing around your chimney.

Possibly dry rot--serpula lacrymans?

From Wikipedia:

Serpula lacrymans has a preference for temperatures of 21 to 22 °C (70 to 72 °F) but can survive any temperature from 3 to 26 °C (37 to 79 °F). It is not clear how much light is needed to promote Serpula lacrymans growth. In terms of aeration Serpula lacrymans often grows near ventilation shafts which shows a preference for concentrated oxygen. A moisture content of 30 to 40 percent is its ideal level in wood to promote fruit body formation.[5] It appears that Serpula lacrymans requires an environment where both inorganic and organic materials are present. The fungus uses calcium and iron ions extracted from plaster, brick, and stone to aid the breakdown of wood,[6] which results in brown rot.