After an inferior alveolar nerve block, the patient cannot close their eye. Which nerve was inadvertently anesthetized?

If the needle is inserted too far posteriorly/deep during an IANB, anesthetic can be deposited into the body of the parotid gland, where branches of the facial nerve (CN VII) travel. This causes a transient facial palsy, including the inability to close the eyelid (orbicularis oculi is facial-nerve innervated). It resolves once the anesthetic wears off; the eye should be protected/patched in the meantime.