Hi,
I have a 6KW generator wired to a Reliant Transfer Switch and the furnace is one of the breakers that is powered. Every circuit runs great, computers, TV, and lights, except the furnace. On Generator power, the furnace gets to the point of igniting the burners, then shuts them down.
It keeps repeating that cycle as long as powered. No fault lights come on the main control board. Is this a power "quality" issue? Is this a grounding issue? The transfer switch was installed by a Master Electrician and the HVAC company says the furnace is working fine. I showed the HVAC technician what happened when on generator power, and he said he suspected a grounding issue.

Any help be appreciated!

Could very well be a grounding issue. If your generator does not have a ground which is connected to the ground in the main panel, that will certainly cause a problem. It could also be a power quality issue. Have your electrician verify the grounding system & check the frequency of the generator if he has that capability.