Do voltage-gated channels in a neuron use ATP
No, they change conformation in response to voltage. Wikipedia has some general discussion of the gating process, as would any basic neuroscience textbook.
The primary energy (ATP) cost of neurotransmission is in establishing ion concentration gradients, via the sodium/potassium ATPase. You can think of this as a special case of secondary active transport, where the energy costs are in establishing a gradient that is then used for a later process which does not cost any direct energy.