Hi Platonides, this is a great idea, but requires much work; we'd probably need to make a crowd-sourcing effort to get at least a small list of such objects. As far as I remember, Emijrp wrote a great essay on the subject last year, so we'll only need to add the list.
I see that the Global Heritage Fund has identified such a list for Syria (see http://ghn.globalheritagefund.com/uploads/documents/document_2107.pdf), so maybe going in the direction of countries currently engaged in a military conflict (Wikipedia provides us with a list at http://en.wikipedia.org/wiki/List_of_ongoing_military_conflicts) might be a choice.
On the other hand, a Google search for "damaged cultural heritage" (in Polish) directed me to an article on the 2012 Northern Italy earthquakes (at http://en.wikipedia.org/wiki/2012_Northern_Italy_earthquakes), so maybe this direction good help us list some other damaged monuments, too.
Just my three grosze.
Thanks, Tomasz