Yes I did final inspection. The bathroom was left unfinished with rough-in for future. After final inspection was complete, bathroom was then completed to get around the by-laws.
We were in the same boat about furnace, but more about the concern of airflow if we connected duct work to new addition. The fix was we build a small furnace room and put a new furnace and duct work for the addition and left the old part of houses duct work and furnace the same.