Let me also comment...
The output is a function of the input voltage. The idealized Op Amp has infinite resistance and infinitesimal capacitance at the input so no current is required to bring it up or sustain a given input voltage. In reality you allow for positive capacitance (the FET gate capacitance e.g.) and you will have a short transient current as the voltage changes. Allow for finite resistance and you have a small steady state current to compensate for this leakage.
But in principle these can be as small as your like within the limits of your resources to build the OpAmp with the desired characteristics. There is no reason (in the classical world) to think current must flow for the OpAmp to read the input voltage and establish a proportionate output.