What brand (Suburban or Atwood) and model of furnace are we talking about? Details are important.
The burner continues to flame after the fan goes off? That is really strange, since the sail switch, which detects sufficient air flow from the fan, should immediately shut off 12v power if that happens. The fan motor itself could quit as it heats up, but the sail switch should still cause a shut down by removing power to the controller. My guess is that the controller board is at fault and fails to turn off the burner after the fan stops.