That is indeed a fascinating question. We should start by acknowledging the complexity of Greek mythology. Often these stories don't have a clear moral like other well-known stories, like Aesop's fables.
Turning to Orpheus, and looking at moral lessons specifically, one complex lesson comes from interacting with the gods. Eurydice died because she objected to divine desire (specifically the lust of Aristaeus). Can we conclude that desire is dangerous? We can definitely conclude that crossing the gods is dangerous. This can be made into a moral lesson when you go further into the story. Orpheus was allowed to descend into the underworld to retrieve his wife with a single condition: he could not look back to see if she was following. In a way, both of the spouses did the same thing: crossed the gods.
Again, Orpheus was not allowed to look back. This means he had to trust. Trust, in the gods and in love, was the rule he had to follow. He crossed the gods and doubted his true love when he looked back. A moral lesson, then, is that trust is essential when dealing with love.
The story of Orpheus is an amazing one, and Ovid, the Roman poet, has made this story enduring for us.
As for a moral lessons, there are two, in the least. I am sure that others will have their own opinions.
First, one point that comes out clearly is the power of love. When Eurydice dies, Orpheus mourns for her. His true love is now gone, but the power of death was not as strong as his love. He bends the will of Hades, and he goes down into the underworld to get her. Hence, one theme is the power of love; love is stronger than death itself.
Second, never turn back but trust. Had Orpheus not turned back to see if Eurydice was behind him, they would have both made it out of the underworld together. But because Orpheus did not believe and turned back, he lost her a second time. Hence, another moral is press on and don't look back. Finish your mission not matter way.