Huh? I am neither an HVAC or copper expert, but this makes no sense to me. Any kind of residue/soot from a furnace I've ever seen has been black or grey. Any kind of rust/oxidation run-off I've ever seen has been red or brown. All signs point to this being related to the flashing, and not the furnace. Maybe not the copper itself, but perhaps some other metal underneath it? I think the homeowner may be in denial about it being the copper, because it's expensive and they like it. Not sure why the HVAC guy thinks it's the old furnace, though. Maybe it helped him sell the HO a new one, lol
As for treatment, maybe F9 if you think it's metal related, or some degreaser if you really think it's furnace related. But go easy on that, since degreasers are inherently bad for asphalt shingles. A lot of times soot will come off with a SH based housewash mix.