Many old devices didn't check for a proper ground or a neutral bond. But as devices have become more complex with sensitive electronics controlling a lot of power, the need for properly bonded and grounded power has become more important.
We are talking about 2 different things here.
1) All metal housings that contain wiring should all be connected to the building grounding conductor back to the electrical panel where the disconnect is, and that should run to a ground rod from the main panel. The dual ground rods that many codes are now requiring, are still to be connected by a single wire to the ground bar in the main electrical panel. There should be no other ground rod in the system. This ground wire path throughout a build should be able to carry the current required to trip the break of the circuit running with it.
2) The neutral wiring going to all outlets and devices all need to connect back to the main panel neutral bar. These are the wires that are intended to carry the return current. If everything is operating correctly, there should be no current returning on the grounding conductors. The neutral should be bonded to the ground system at the first disconnect into the building. There should never be another point in the system where the neutral and ground connect.
Ground fault interrupters work by comparing the current on the hot wire to the neutral wire. As long as all of the current going out the hot wire is coming back on the neutral wire, it knows all is good and leaves the power on. If the balance between the two currents differs by just 60 milliamps, the device will shut off the power. A full floating power source such as a generator on rubber tires with no ground will not trip a ground fault unit because the current can only flow between it's hot and neutral. No neutral/ground bond, no ground fault detecting. A furnace with a microprocessor control board, a CNC machine, or an EV charger all need to see clean power with proper grounding. If they see a voltage potential between the neutral and ground, many of them will shut down to protect themselves as well as people that might touch the device. This is for equipment protection as well as human protection. An EV charger is a delicate device, but controlling many amps of current. Same with a CNC machine and the gas furnace. Any current on the chassis ground could destroy the electronics as well as kill people. The electronics test the grounding and may now power up if they don't detect a proper hot, neutral, and ground system. Many electric cars won't charge from a generator unless the neutral is bonded to ground, and it has a path to earth ground from a ground rod. Ran into that problem trying to charge a Nissan Leaf with my old 5,000 watt Coleman generator. It kept showing a power fault until I bonded the neutral and we used battery cables to jump the generator ground to a speed limit sign post. Bing, the car started to charge. Took an hour to get 8 miles of range into the car. Should have been faster, but we could only get it up to 2,000 watts of charge power before it would fault again. I think the generator frequency or voltage was changing too much at higher load. That was enough to get him so a charging station.
My guess is that the furnace is also checking the power system to ensure it is safe to open a gas valve and light a rather intense fire under the control of a small computer. My 10 year old Carrier furnace is microprocessor controlled. The start up sequence is pretty picky and it can set about a dozen different fault codes and not light up if anything is wrong. It actually measures the electrical impedance across 2 wires in the flame to determine if it is actually burning or not. I can see this failing to work if the unit is not properly grounded.
With the inverter having the 60-0-60 output, you can't just bond the -60 volt "neutral" to ground. If all of the electronics in the inverter are fully isolated from the case, you "MIGHT" be able to then connect that "neutral" to the neutral wiring of your house, but it is unlikely the battery side will be isolated. Even if the inverter case is floating from the battery wiring and the output leads, you may not be able to safely ground the inverter. Unless the inverter is fully isolated, which I highly doubt with a 60-0-60 output, then the battery terminals of the inverter would now be at 60 volts, with the output now being 0-60-120 volts. That is not safe. This puts the battery powering the inverter at 60 volts AC from ground and that could kill someone. An isolation transformer will fix this problem, or you may be able to get an inverter with a proper neutral for just a little more money.
When feeding a device in your home from a generator or inverter system, you should leave the neutral and ground wires still connected back to the main panel. Connect the neutral and ground of the backup power source to the house neutral and ground, and then the new hot wire from the inverter or generator to the hot input of the device. This is how a generator transfer switch panel works. I installed one similar to this at my old house for use with my Coleman genny.
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I had my furnace, refrigerator, microwave, a PC, some lights, and a TV on those circuits. Plug in the genny, start it up, and flip a few switches and I am back running during a power outage. But at my house now, I have a Schneider XW-Pro battery inverter feeding a sub panel for my essential loads. It is completely automatic if power fails it just starts inverting and runs the important loads in my home. And my solar can charge it when the sun is up.