I am certain that if there was a perfect replica of the Mona Lisa, the original would retain most of its value. I don't really understand why this is the case, but I do understand that it is the case.
The value for me is in knowing that I am standing in front of the same canvas that Da Vinci stood in front of. It's similar to going to a historical site and standing where history you've heard about was made and seeing/hearing/smelling something of what the people who made it did. It builds a sense of human connection, a shared experience, and makes something you "know" to be real "feel" more real.
Da Vinci never stood in front of the "atom for atom copy" of the Mona Lisa, and generations have not come from around the world to see it. We're receiving the same visual stimulus, but the ability to participate in the human connection is reduced because we know it's a copy. (Worth noting that if we didn't know it was a copy, then this wouldn't be a problem.)
Interestingly, when paintings like the Mona Lisa are concealed in those plastic protective containers, that experience is significantly decreased for me. It's strange, but that layer between the canvas and my eyeball has a major effect on my perception.