My understanding is that the clerics fear idolatry. They are concerned that people are worshipping sites like Mohammed's grave and his mother's grave when they should only be worshipping Allah. It's a bit like the concern that many Catholics are worshipping Mary instead of God. So if they believe people are worshipping a site instead of Allah they're inclined to move to tear that site down.
Big danger of drifting into religious discussion here, but I believe that is the motivation and also why ancient pre-Islamic sites are not of such a concern so long as people are no longer participating in any religions they may represent. For example access to the ruins of a recently discovered ancient(ish) Christian church has been blocked, though I believe it hasn't been destroyed.
That is my understanding of the religious motivation as well. Typically speaking, they value the pre-Islamic sites because they see them as connected to the history of their people as described in the Injil, Tarwar, and Zabat (their versions of the Jewish and Christian Bible). This is to be remembered in their mind, and the prophets honored. Anything after Mohammed is not of the same value, and has the possibility of being honored to the same level as the prophets. Hence, Islamic Era sites are more likely to be destroyed in some interpretations of Islam than other sites.
This is one of the many things, btw, which set ISIS apart from other interpretations of the Quaran. Instead of honoring historical sites from before the time of Mohammed or studying the classics, they believed that any such sites could detract from veneration of Mohammed, and more or less rejected the Injil, Tarwar, and Zabat.
Now, on the political side, it wouldn't surprise me if they suppressed knowledge of especially Jewish sites (they may fear Israel may try to claim them as their territory, and that would be very bad for all parties involved), or if they genuinely consider them Islamic sites (Islam believes it originates from Abraham as well).
As an aside, I see this as more of a sociological and historical discussion which happens to include descriptions of what is believed by a group, rather than a discussion of religion itself. As long as we can stick to that, hopefully, this discussion can continue...