If it is indeed gas fueled, have the pressure and regulator checked. How many gas appliances are in your home? Proper size gas pipe to your furnace is required so it’s not starved from other gas appliances. Sometimes the “whistling” noise you described may be due to undersized piping and/or pressure. Your local utility may help you out. Heck give them a call, you pay the bill every month.

Turn the thermostat up, so the furnace cuts-in.

Go to the furnace location, listen & see what is happening. I’ve seen older gas furnaces that had cast iron burners in them, when condition where right, when furnace was off for a while, and when they lit, there would be this wierd wooing sound comming from the burners. After a few moments when the furnace heated up (the cast burners also heated up and this horrible noise would stop).

Vaccuum the lint around the venturi tubes.

What I did next was turned the air shutters on the venturi tubes on the burners slightly one way or the other. It’s a combination of fuel and air ratio. As plumber Tom said in his comment it could be gas pressure. Is this furnace Natural gas or propane?

Again you have to spend in little time and find the origin of this noise.

Do you have a old counter balance type humidifier on the furnace? You know the style having a saddle valve on the water line & 1/4″ copper tubing going into the humidifier that had 5 or so pads to soak-up the water and a pan. As the pan filled the weight of the water would shut off the water supply by putting weight against a nozzle.

I’ve seen these humdifier lines vibrite back or forth making a noise.

It can drive you. Again spend a little time looking for the source. Perhaps you should call the gas company on the burner adjustment. If noise is from the burner area. You may get it to rich and carbon up the furnace, and make matters worse.