Rumbling noise from the furnace when it kicks in and while it's running

Sounds like a bad bearing in the motor or blower cage assembly, perhaps a support bearing if so equipped. In some older models, a loose fanbelt or motor will cause this sound.

I am going to guess that you're combustion chamber motor bearing is failing. I'm on my 2nd motor on a 4-year old furnace. I'm suspecting a design flaw. I won't be surprised if I have to change this motor in 4 more years again.