One could argue that the overriding moral lesson one learns from the story of Orpheus and Eurydice is that there is a moral law higher than ourselves. One doesn't have to believe in the gods of Mount Olympus to accept this; to some extent, all forms of morality entail subjecting ourselves to a set of rules that we must follow for the good of ourselves and others.
In the ancient Greek myth, the relevant moral rules don't demand all that much of Orpheus. All he has to do is emerge from the Underworld with his beloved Eurydice without turning around to look at her before they reach the light.
But because Orpheus can't hear Eurydice's footsteps behind him, he assumes he's being tricked by Hades, the god of the Underworld. Unable to trust Hades, and by extension submit himself to a morality higher than himself, Orpheus makes the catastrophic mistake of turning around to see if Eurydice has followed him. When he does so, she vanishes into thin air, and her shade returns to the Underworld, where it will remain for all eternity.
In looking back to see Eurydice, Orpheus was effectively disrespecting the gods, showing that he didn't trust them. More significantly, perhaps, he was showing his impatience with the necessity to submit himself to a moral code to which continued obedience was ultimately for his own good.