My furnace makes a rumbling noise when the blower begins. Does this mean that the blower or furnace motor needs to oiled?

Usually when that noise is so pronounced the motor bearings are shot and the motor will need to be replaced. The days of lubricating most motors are almost over since most of them are sealed units and you cannot oil them without a lot of trouble.

The blower wheel might have also come loose or something might have been caught in the blower wheel like the wiring diagram that is on the back side of the blower motor access door.