With the unit in heat mode and auto, the burner ignites but the blower doesn't come on.

Could be a defective fan and limit switch or the heat fan relay.

Or a bad blower motor, or if the unit is very old, it could be a broken fan belt. Can you give us a make and model number of unit? It could also be as simple as a bad blower motor capacitor. Just to many options without knowing what unit you have.