Is that what UNESCO said?
In 2017, UNESCO declared the old city of Hebron to be a "Palestinian heritage site." But its description of the Cave of the Patriarchs does not say that it was an Islamic holy site alone:
The main monument of the town is the centrally sited Al Haram Al-Ibrahimi Mosque/The Tomb of the Patriarchs. Elements of the current building date back to long before the Mamluk Period, as do its religious associations and the reasons why it is revered by Christians, Jews and Muslims alike. The mosque is said to host the remains of God’s prophet Abraham/ Ibrahim, his wife Sara, their sons Isaac and Jacob and their wives Rebecca and Leah, as well as Jacob’s son Joseph. (?)
There is reference in the Book of Genesis to Abraham purchasing the field for the tomb. The sanctity of the tomb site was known from as early as the Herodian Period, (1st century BCE) when a monumental enclosure was built around the sacred Cave of Machpelah, whose location is now lost. This enclosure of massive, finely dressed stone blocks still frames the mosque and within it are structures that reflect later Fatimid, Crusader, Ayyubid, Mamluk and Ottoman periods. The great covered prayer hall was constructed in 12th century out of the remains of the Crusaders 11th century Romanesque church which in turn arose from the ruins of a 7th century mosque.
Habbash is lying - UNESCO never declared Hebron or the Tomb of the Patriarchs to be "Islamic," let alone exclusively Islamic. It recognizes that that main frame of the site was built in Herodian times, under Jewish self-rule. It also recognizes that it was a holy Jewish (and Christian) site way before Islam.
UNESCO simply declared it Palestinian based on the world believing that it is part of a mythical nation that never existed. But it never said a word about it being exclusively Muslim.
Given that under international law, people have the right to visit their holy sites, Palestinians are lying when they try to ban Jews from worship in sites that were holy 2000 years before Mohammed ever soiled his first diaper.
I also will note that already in 2009, when "Palestine" first bid to join UNESCO, they made it clear that the primary purpose of them joining that organization was to ban Jews from holy sites.