I've got two Honda generators, the Eu2000 (inverter)and the EP2500cx(conventional brush type)The EU2000 is a great little generator and I use it 90% of the time due to its portability but it has it's limitations?Could it run a furnace and fridge? I would say probably as I've done it before but it would struggle on startup.The EP2500 definitely has more punch for starting heavier equipment ,for instance it will start up a small compressor that I have but the EU2000 won't.The EP2500 will easily run a furnace and fridge and still be able to do some lighting.Its a heavy little mo fo but will run much longer on a tank of gas than the Eu2000(8 hours as opposed to 4 1/2 loaded up).Yes the inverter is a better choice for electronics but with a decent conventional generator you probably will never have a problem with circuit boards etc.I paid $799 for the EP2500 and $1299 for the Eu2000.Hope that helps.Basically the Eu2000 is great but the cheaper EP2500 makes a better workhorse.Throw a sump pump into the equation and the choice becomes clear.
I have a similar set and think the 2500 conventional has more brute force. Pure speculation is that Honda protects the EU types with quick trip breakers to protect the electronics.
Going to the original question I see two concerns.
1) Will the generator run the furnace?
2) Will the furnace develop any problems due to circuit fluctuations?
Question 1 can be resolved with a five minute test. Question 2 is the trickier one. What good is a generator if a circuit board or motor gets fried in the furnace? Sometimes the failure is instant but in other cases it takes repeated surges / spikes to do the dirty deed.
I'm pretty sure the furnace makers would not officially approve powering with a generator. Why should they? The variables in generator types and user competence are too great.