Syria is host to 5 other World Heritage Sites, all of which are now considered "in danger". Yet, despite the Islamic states' willingness to destroy these sites, no additional protections have been implemented --- and this is to be expected. For those who read The Monuments Men by Robert Edsel (read the book, it's far better than the movie), it comes as no surprise that cultural and environmental protection are among the first principles sacrificed during conflict. Yet, the lack of active response is just as much a result of the design of the institution intended to protect these sites, as it is the nature of war.
The World Heritage Convention, signed in 1972, was intended to provide structure and encouragement for states to protect sites of universal cultural and environmental value --- which has currently been signed by 191 member of the UN. The primary objectives include conservation, credible protections, capacity-building, open communication about protection efforts, and incorporation of local communities in conservation efforts. It established the World Heritage Committee within UNESCO, as well as the World Heritage List --- a list of sites determined to be of value to all humanity.
Although the program is intended to improve protection, it is like other mutilateral agreement --- a result of negotiation among member states, a relies primarily on the activities of member states. For example, in order for a site to be added to the World Heritage List, it must be nominated by the country within which the site lies, and must be approved/voted as of universal value and adequately protected by the World Heritage Committee. Once on the World Heritage List, the host country is responsible for its administration and protection. The only recourse for improper protection and destruction is for it to be delisted --- requiring another vote by the World Heritage Committee.
This design was essential to gaining participation by member states --- few states would be willing to sign on to a convention that allowed other states to nominate sites and then either required the host country to invest in its protection or allowed other countries to be responsible for its administration and protection. Thus, it is a design necessity. However, it also fails to provide for the protection of important sites when the host country is unwilling or unable to protect them.
The tradeoff between participation and effectiveness is a common one in international institutional design (among many others). It is also one of the primary forces that drives the creation of ad hoc methods of improving effectiveness. For example, under some conditions, countries can obtain additional funds to administer and protect sites through the World Heritage Fund. In light of the destruction of cultural sites and artifacts by the Islamic State, many countries have implemented additional efforts and funding to help with protection and recovery. UNESCO has led a variety of coordination meetings. And a variety of NGOs, including the Syrian Cultural Heritage Initatives have joined forces with UNESCO and member states to document damage and plan for recovery. But, to date, the international response is largely about post-conflict recovery. The actual protection remains in the hands of the state government.