Not sure if this has been kicked around yet. I have several customers with portable generators that can't run their furnaces. Nearest thing I can figure is that they are not putting out 60 hertz. I also tested several that had an open ground at the GFCI receptacle and L14/30 receptacle. So 2 questions: Is there a simple way to get the output hertz up to 60 if the generator is only putting out 57 or 58.
And separate from the above question Why is there no ground on the L14/30 receptacle of certain generators.
There are $99 generators, and there are $1300 generators - even at the same modest "20 amp / 120v" output range. Ever wonder why the price range?
One of the big areas where they 'save' expense in in the power regulation. The cheap ones are not putting out a very good waveform. This is just fine for lights and motors, but can be pure nightmare for anything electronic - like modern furnace controls.
Other drawbacks to the cheap genny's are high fuel consumption and high noise levels.
And, as Trumnpy has mentioned, overloading the genny will result in the voltage gropping before the machine simply stalls.
A furnace should only have about 300-500W running the fans/oil pump at 120v - and with an induction motor, frequency should not be a problem, unless it's so low it causes motor overheating, [which is not immediate anyway]. It may be under or over voltage rather than frequency - many modern electronic burner controllers have voltage protection both sides of the norm and may also baulk at square waves or spikes. Her's the Danfoss spec - http://heating.danfoss.com/PCMPDF/OBC_82-10_X009114_VDACY202.pdf
First, do not assume all are caused by the same problem. Many cheaper generators have poor regulation. I have seen many that will show 120v but low hz when you raise the rpm to get 60hz the peak to peak wave form will exceed what the equipment will handle. Second issue is grounding. Newer furnaces use flame detectors (Flame Rods) that much rely on grounding. Power with a floating ground will cause the CPU to detect no flame and shut down. I have a Honda EU200 inverter generator, it will not operate my furnace unless I ground the neutral side (It has no internal ground) and make sure all including the furnace and gas line is all bonded together.