If it would start every time you unlock the car, like he mentioned, you bet an extra 0.5l/100km would come up on your OBC.
As about the faulty heater unit, yes, it stores fault codes, but they are more related to failures to start the heater. If you don't have frequent temperatures below 7.5 celsius (if I remember correctly), I would rather disable it then pay the labor and parts for fixing it, which usually are quite expensive.