The description of arcane eye states that the eye created by the spell "can't enter another plane of existence." PHB p. 214.
Consider this scenario: a warlock casts arcane eye (perhaps using the Visions of Distant Realms invocation), creating an eye 30 feet away; subsequently casts demiplane; and then steps through the shadowy door created by demiplane, leaving behind the eye exactly where it was created.
What result? Is there any rules-as-written reason to rule otherwise than that the eye did not "enter" another plane, and so remains in existence and continues to transmit information across the planes? Or does the eye remain in existence but cease to transmit information? Or does it cease to exist? Or something else?