The newest Vice documentary The Islamic State vs Lebanon brought up a tragedy that often gets buried in the countless other issues going on in the Middle East: priceless historic relics at the hands of people who will definitely destroy them without hesitation. Is there a group that specializes in securing these relics? Whether it's smuggling texts or artifacts out or securing a site.
Yes, the UNESCO World Heritage Center (under the UN) monitors historic areas that are in danger from natural or political threats and provides assistance. Sometimes the guardians of historic items will smuggle things around of their own accord, such as with the Timbuktu librarians about a year and a half ago. But obviously people from more powerful countries going in to troubled countries and pillaging historic treasures "for their own good" is not something any reputable historical group is going to be doing.
Yes. In addition too /u/caffarelli's post there are people working to preserve the art and artifacts of Syria: http://www.spiegel.de/international/world/how-archaelogists-are-trying-to-save-syrian-artifacts-a-983818.html
Unfortunately even in relatively stable countries with dedicated and functioning antiquities bureaus the looters can still get in: http://www.washingtonpost.com/world/middle_east/looters-raid-jordanian-crypts-in-search-of-gold-jewels-and-artifacts/2014/10/29/67a53b46-5ac7-11e4-8264-deed989ae9a2_story.html