It would ensure no greater protection and would probably hand Ani permanently over to those who are currently misusing the site and destroying its cultural importance. UNESCO is a political body - that was clear after its extreme reluctance to make any comments or take any action on the issue of the Djugha cemetery destruction - so I do not think it as an organisation would be capable of protesting against damaging activities taking place at Ani or prevent such activities.
In addition, thanks to the destruction of Ani's integrity as a historical site (caused by recent extensive and heavy-handed restoration works and rebuilding), I doubt that Ani can genuinely now meet the standards required to be listed as a World Heritage Site. If it were to be listed, it would just be additional proof that UNESCO is all about politics.
I had heard a site visit was going to be undertaken by some organisation, don't know which (so maybe it was ICOMOS) in September - but that it was cancelled as a result of the current internal unrest within Turkey.
I have been intending to write an article on the World Monuments Fund and Ani - it would be highly critical of its (and its fellow "stakeholders") activities there and of the ideologies that have been expressed in relation to the site. It would be an expansion of the brief response I made herehttp://armenianweekly.com/2015/01/29/ani/