Hvac Fan only blows in 'fan on' mode while heating
Normally, when the fan is set to "auto", if the unit enters "heating" mode the burner ignites then a heat sensitive device sends a signal to a relay which turns on the fan. The fan pushes heated air into the plenum and through the ducts but it also removes heat from the combustion chamber, so if the fan does not start or fails during operation, the burner shuts off to prevent overheating, damage, fire, etc.

If your fan is not coming on it could be a bad heat sensing device or a bad relay (the fan motor and power supply seem fine as you state that it works under other circumstances).

You did not state whether or not the burner continues to operate without the fan running, if this is the case you may have an additional problem if your system is designed (as most are) to shut down the burner upon fan failure.