Thanks everybody for your replies. I wasn't brave enough to replace
the circuit board on my own and my neighbor was able refer a reliable
serviceman. He did some tests. Namely, he took out the high-limit
sensor and used a wire to bypass it. It turned out that was the
problem. The faceplate on the sensor was a slightly warped so it was
bent to the side and touching the side of the heat exchanger (or at
least too close to it), and thereby continually triggering the
sensor.

The repair guy just re-installed the same high-limit sensor but
without screwing it in so tightly, and so it wouldn't be . This was a
temporary fix until I can order a new sensor and replace it. (I'm not
sure if I explained this exactly correct, but it's fixed and I'm
happy.) Thanks again everyone!