Final answer:
The orbicularis oculi muscle is controlled by the facial nerve (CN VII), not by CN III, CN IV, or CN VI, which control the extraocular muscles.
Step-by-step explanation:
The muscle responsible for closing the eye, the orbicularis oculi, is controlled by the facial nerve, which is the seventh cranial nerve (CN VII). The oculomotor nerve (CN III), trochlear nerve (CN IV), and abducens nerve (CN VI) are responsible for controlling the extraocular muscles, but it is the facial nerve that innervates the muscles of facial expression, including the orbicularis oculi muscle. If a patient cannot close their eye due to orbicularis oculi weakness, this may indicate a problem with the facial nerve.