Can non-metallic cable go up a wall shaft for a furnace vent?
The first test would be to verify the area is not hotter than the wiring can handle. Romex is rated for 90 deg C or 194F.  The next check would be is the area being used as a plenum for the fresh air supply (some older homes used this airspace for environmental make up air ). If it is a plenum, romex is not plenum rated and may not pass an inspection if noticed (would need pipe) If just a hollow space romex would be ok as long as not two hot. EMT really would not help as far as temp is concerned and would limit the number of wires in the EMT or need to be derate based on the total current carying conductors.