This problem sounds like a flame sensor. Most modern furnaces have several safety sensors and switches. What you describe sounds like a flame sensor. Here is how it works. When the burner starts up the gas valve has a timer that was a few seconds for a circuit to close from a small bimetal rod that is in the flame. If the furnace does not light the flame sensor doesn’t heat up and the loop is not closed. In this state the gas valve shuts down so the house doesn’t fill with gas. The problem is that you can see flame so the flame sensor is sending the wrong message to the as valve. Often the flame sensor can be cleaned and it will work fine again. If you recently created a lot of dust such as dry walling your basement that will usually create this problem. Impurities in the fuel also cause the flame sensor to get dirty and fail.