Furnace control board automatically checks and detects for open neutral to equipment ground and should automatically lockout the furnace from operating. Hence the reason why you get no response from the furnace.
Some control boards actually have a diagnostic LED on board which will blink out a trouble code to help pin down the issue..
Assuming you have a Honda 2000i inverter series, as I recall those generators do not connect the neutral position to the equipment ground. This typically is not a problem with most electrical items. But it does create issues with things like RV EMS systems or in your case a control board that monitors the neutral to equipment ground and with no connection it faults on the open neutral/ground.
As I stated, the control board uses what is called diode rectification to determine if flame is present via a metal probe which is isolated from the burner and when a flame is present the flame engulfs the probe.
The flame conducts electricity in one direction (like a diode) between the rod and burner.
This sensing only works correctly when the burner is at the same voltage potential as the neutral. For that reason the control board will prevent the furnace from starting or responding to the T stat call for heat.
It is part of a safety system to prevent gas valve from staying open if no flame is present.
Retry your setup with adding the neutral/ground plug on the gen, it should work once you establish that connection.
