SY8113 based Buck Converter Voltage Regulator design
What is the consequence of designing such a power supply circuit for
  3A, and then only pulling 250 - 500 mA from it?


The data sheet says: -


3A output current capability


This means it is capable of delivering 3 amps. If your load is 250 mA then there is no problem. There is also a typical circuit on the front page that shows efficiency vs load current: -



As you can see, the load current range if from 10 mA to 3 amp.

In addition the regulator is a synchronous type and this means that it will handle a large range of load currents without excessive ripple voltage on the output i.e. it won't go into burst mode unexpectedly.


  Secondly, if I calculate the Inductor value from the datasheet
  formula, I get a minimum inductor value of 11.22 uH


You don't need to - stick with the 2.2 uH design and you'll be OK.