Why does an op amp need to be powered before applying the input signal?
Almost all ICs are designed with the assumption that the input and (usually) output voltages will remain between the power supply inputs at all times.  To ensure this, most ICs have protection diodes between the input pins and the power suppy and ground terminals.  

If you apply power to an input when the IC not powered, these protection diodes will conduct, and attempt to power the IC - and any othter ICs connected to the supply rails.  This will cause current to flow along paths that the designer didn't intend, and may cause damage to the IC.