Heels hurt at night? Can shoes cause tunnel syndrome?

No. Shoes themselves do not cause tarsal tunnel.
Something else? Your heel may be hurting because of plantar fasciitis- an inflammation of the tissue that forms the arch of the foot. If so your heel will be exquisitely tender if you press on the sole of your foot where the arch comes off the heel. If so, a specialized sock that keeps the arch stretched at night can be helpful.