my furnace turns on then shuts off after a few minutes after about 15 seconds the igniter lights furnace starts and repeats itself 3 or 4 times then does not turn on. If I turn the thermostat off then on it will light up and and do the same thing.
Furnaces shut down when they don't sense flame after several attempts. Turning the stat down then back up resets the safety. If it has a flame sensor rod clean it and the burners. Lots of the older rheems used the hot surface ignitor instead of a dedicated flame sensor so yours might not have one. New updated control board and sensor kit is more then you want to spend on a 20 year old furnace.