How do you adjudicate something like Hammurabi’s Stele (Code of Hammurabi), then? It was created by a culture which existed within the borders of modern Iraq, but was found in Iran. To whom does it “belong”? Especially given that unlike the Greeks or Indians today, modern Iraq and Iran have basically no connection to the cultures of ancient Mesopotamia despite the overlapping borders.
That’s a tricky question.
I’m not so convinced that you can just declare that a country has “basically no connection” to an ancient culture that occupied that space, since there’s very little that modern France has in common with, say, the Romanized Gauls of the 1st century AD, but arguing that because of the cultural shifts that have happened since then the Pont du Gard should be repatriated to Italy is insane.

On the other hand, one of the biggest problems with nationalism is that a lot of nationalist arguments are particularly sweeping and essentializing – X defines our in-group, all of us are like that, we’ve always been here, etc. – and run into huge problems when you have competing nationalisms with equally sweeping and essentializing claims, as in the real world there are rarely clear boundaries between national groups.
At the same time, I think there’s a pretty strong argument that the connection between Hammurabi and modern-day France is significantly more attenuated than a connection between the Stele and either Iraq or Iran, and that Gustave Jéquier wouldn’t have been allowed to remove this antiquity under modern UN conventions.
So who should get to keep it? Well, the Stele was taken to Iran in the 12th century BC and Cyrus the Great of the Achaemenid Empire (which included both modern Iraq and Iran) put copies of it throughout the Empire, so it’s hard to argue that it doesn’t have Persian as well as Sumerian cultural significance. If it were up to me, I’d say that it should be shared between the two countries and rotate between them, but that’s probably naive of me.