"Hello: I'd start with the simple approach of asking the neighbor to refrain from dampering down the fire so it can burn hotter and cleaner. If they refuse to change habits, I'd ask the local jurisdiction (city or county) if…"
"Hello: If it's an FVIR heater, there can be restriction in the air intake caused by nearly invisible dust. Washing off (or blowing off, or vacuuming) the intake screen and flame arrestor might not be a bad thing to do. …"