Multiple induction coils in series (phone charging)
Does there need to be more energy flowing through the circuit as the
  number of coils increases? (i.e. from 1 to 16)  


Yes, a coil or inductor simply transfers current to magnetic energy, because of conservation of energy, you'll need more energy in the source coil if you have several load coils.


Source: Electronics WS transformers


  If multiple devices are being charged at once, does there also need to
  be more electricity flowing through the circuit?


You'll need more current, yes. 


  Will the coils in the array interfere with each other?
    Depends on how they are arranged, the 'source' coil will source a magnetic field, if a receiver coil attenuates the field of a coil adjacent to it then you will have a reduction. Look at magnetic circuits, calculate the magnetic field from your source coil.



Source: WPC