Does electrical usage contribute to fuel consumption?
Short answer

In flight, unless you use external energy sources like the Sun, each change in the use of electricity leads to a proportional change in the quantity of fuel burnt in the engines (or in the APU).

On an aircraft, generators only convert fuel potential chemical energy into something else, usually electricity, hydraulic energy or pneumatic energy. 

There is no other source of energy:


The batteries themselves need to be charged, charging burns fuel.
The RAT (ram air turbine) uses energy from the relative wind created by the engines thrust. If the RAT is used because the engines and the APU are down, the relative wind is due to the loss of altitude created by the gravity. This kinetic energy is the restitution of the potential energy accumulated during the climb with the engines, so with fuel.




Details

Maximum power of the electrical equipment

The electrical power needs of the whole aircraft can be completely met by a single engine generator, or by the APU. On the A320 family, the APU is an APS3200 delivering about 90 kVA.

APU fuel consumption

From this PPRuNE discussion, an A320 APU may consume 130 kg per hour on the ground, and 51 kg per hour at FL300. However the APU provides at the same time electricity and bleed air energy.

APU fuel consumption for electrical energy generation

However the fuel quantity used for the sole electricity generation may be approximated:


Maximum power required by the aircraft equipment → 90 kVA AC.
Assuming a mean cos φ = 0.8 → Apparent power = 110 kW DC.
Assuming 25% efficiency for the system (gas turbine + generator) → Fuel power required 440 kW.
Fuel energy required → 440 kWh per hour.
Converting to joules (1 MJ = 0.28 kWh) → 440 kWh ≈ 1,570 MJ.
Kerosene specific energy = 43 MJ/kg → Fuel quantity burnt in an hour = 1,570 / 43 = 36 kg (45 L)


So approximately 36 kg of fuel are required per hour to provide for electric energy on an A320.

Note that in flight all engines generators will be active and will share the total demand (feeding different buses for safety), and the APU will likely be inactive. The fuel consumption should be more or less the same, regardless of the number and nature of the active generators. An inactive or unloaded engine generator can be seen as not impacting the fuel consumption for practical purposes.

Engines burning the maximum rate of fuel

Engines have a maximal power. When the engines are already at full thrust, increasing the electric load will divert an additional amount of energy to the generator. This quantity being removed from the energy usable for aircraft propulsion, the aircraft will slow down.

(Updated, taking into account the multiple helpful comments -- Thanks!)



Bonus

Breathing increases fuel use too:


Each time a passenger breathes, they reject heat and CO2 which must be evacuated out of the aircraft.
Some quantity of air must be replaced by new air by the conditioning system.
This system uses pneumatic or electric energy, and again fuel must be burnt to create this energy.