We like to have our Trane furnace fan on all of the time to keep the air circulating in our house. Lately the fan is making a high-pitched whine or hum that is driving me crazy. Can it be oiled or how do I fix it?

Is it a belt driven fan or directly on the motor?

I think it is motor driven. It is a 5 year old furnace---I'll try & find out

Probably because you run it all the time its worn out,, most/near none can be oiled,they are packed from the factory,unless the squirrel cage is rubbing the housing or its loose and the blades are spinning on the motor shaft, its probably warning you its going bad

I have ran my blower 24/7 for the last 6 years. They are made to run. But it is not uncommon for the hub to break loose from the wheel or for it to get out of balance and start to rub(not from running all the time). A bad run capacitor will give a motor noise. I would get it checked out.