At this point in time I can only repair convection blower harnesses and connectors... but nothing on this board is particularly advanced hardware wise other than the fact that it is 4 layer and has extensive ground planes obscuring many traces. I can definitely take a look at it, but can't guarantee it will be repairable until I find out exactly what's wrong.
The control circuit for the ignitor appears to be fairly simple (C266, D34, R257, R259, D43, Q87/86, RLY1, single GPIO on the MCU driving all of it, appears to be a standard NPN ground side switch circuit driving the relay winding) but I have not spent more than a few minutes looking at that section yet. Looking at it my first guess would be that either the contacts are welded on the relay or Q87 and/or Q86 is shorted. Both appear to be SOT23-3 HABT2222/2N2222 types.
To truly put these through their paces and verify before return shipping, I would need a test unit furnace locally to use, but if you can describe the exact problem (as you just did) it is very likely I can repair without that. I would be much more comfortable with having a test furnace, naturally. I do not like unknowns.
EDIT TO ADD: Yes, after spending a few minutes more looking at this. C266 is a filter capacitor, it is fed from the power supply input connector at the top center of the board. One diode blocks reverse bias of the power supply being hooked up backwards from forcing the relay on; the other is a reverse EMF snubbing diode for the relay winding. R259 is a current limiting resistor that they have used as a standard design feature in every 12-15VDC powered subcircuit on this board. R257 (I might have the two resistors swapped, I already screwed the cover back on and I'm not taking it off again at this point) is a current limiting resistor that limits current from the MCU GPIO pin into the Darlington pair base. Q87 and Q86 form a Darlington pair that grounds the negative side of the relay to turn it on. As long as the microcontroller GPIO pin itself is not damaged I can almost certainly repair this failure. I would say 90-95% chance. The only catch is that the relay is discontinued on all my sources so I will have to find an appropriate substitute with the same footprint and equal or better specifications, so hopefully it is the driver transistor(s). I would guess 75% chance of the transistor further into the corner of the board being shorted C-E and 25% chance of contacts welded in the relay, just looking at the overall layout of this circuit and your failure description.