are these or any other indicator lights on the PC board or display ON?
So far it sounds like a flame sensing problem this can happen if the furnace is not properly grounded or in older water heaters where the ground connections might be corroded.
You can do a heat stress test on the control board.
Use a hair drier to slowly and gently warm the card to 120F(warm to the touch but not hot and not over140F)... if you get any change in behavior that means the card is bad...
A micro crack has opened in the printed circuit at a solder joint, or inside its CPU in that case .... or a short circuit has developed as the card expanded when you heated it.
Constant expansion and contraction as these electronics heat and during their duty cycles over the years is what wears these cards out.
It is also a good idea to replug all of the connections in the unit,that burnishes any oxidized terminal connections and will sometimes solve the problem, especially at the printed circuit card and in any sensor molex connectors. Do that with the power off to the unit of course.
Let me know what you can discover, we can go from there.