I have a new, 3-year old, high-efficiency furnace in my home. Last year it started making a slight whining noise that was audible during the very first stage, before the blower motor started. No problems. Some weeks ago I started to hear a rumbling noise when the furnace kicks on--again, first stage, it's not the blower. The rumbling sounds a bit like the noise my fridge fan makes when its a little off balance. What might be causing this?
I'm assuming since you mentioned a blower motor that you have forced hot air. Your system has two thermostats -- the one you use to set the room temperature controls the burner, and a separate thermostat internal to the furnace controls the blower fan, switching it on when the burner has warmed things up enough to not make a cold draft, and back off when things have cooled down after the burner has been shut off.
You are probably hearing the burner -- if oil, there is a motor, which may have a small vibration; if gas, you could be hearing the combustion air being sucked in, or possibly the solenoid that controls the gas valve.
I have the same situation with a relatively new Lennox furnace (this came with the house, which was built new in 2008). The furnace ran OK for all of two years, and now: every time the thermostat calls for heat, there's this rumbling / vibrating noise. I've isolated it to the cowl that covers the draft inducer motor.
In my case, the only option is to put in a new inducer motor - your mileage may vary.