What happens if you drive the output of a LM78CT05 when the input is not connected
It seems rather unlikely to me that this is the cause. The output voltage has to be greater than the reverse breakdown of the B-E junctions in the regulator to cause damage, and 5V is not. There also has to be significant current flowing back (both conditions are necessary for damage to occur). 

For example, the TI datasheet specifically says: 



I suggest proceeding with caution and looking for something else that may have caused what appears to be significant damage. For example, reversed power connections.