Call for heat on the furnace and the exhaust fan kicks in, the burners light, but the blower fan doesn't come on. What could be the problem?

Have you checked the motor and capacitor? Could also be the relay

If the burner only comes on 5-10 seconds before going off, it could be the flame detector. It is part of the burner assembly where it is exposed to the flame. Find it and clean it.

If the burner comes on a minute or 2, then it likely is the blower controls or even the blower motor. Try turning the fan switch from auto to on. That bypasses most of the components of the furnace fan. If the fan runs, the furnace should run. Once the house warms up, you can think straighter about finding the problem with the motor or controls. Chances are, the relay done it.

If the fan doesn't run, try to see if power is getting to the line side of the relays, and the neutral of the fan is connected to the house wire.