I visited Gaziantep and Diyarbakir last year. Fortunately, most people I still have contact with are fine (depressed, scared, most of them sleeping outside in the winter cold, but alive and fine !).

I tried to get information about the WHS in the area, but did not find much information so far (as Jurre said the attention must go first to saving lives).
UNESCO issued a statement today (https://whc.unesco.org/en/news/2521)
On what I could find in the news and on my recollection of those sites, here is my estimation of the potential damage:

Diyarbakir fortress and Hevsel Garden: major damage have been reported to the old city (many fragile momunent but outside core zone), but no information on the fortress wall or the Ongulzu bridge so far: major damage still to be expected.
Göbekli Tepe: near Sanliurfa city, where major damage have been reported (including a video of an appartment building collpase on the second earthquake): no information on Göbekli Tepe so far. Some of the pillar seem very fragile: some damage expected
Nemrut Dag: very close to the fault line; major damage in the nearby city of Adiyaman. At this time of the year, Nemrut Dag is under heavy snow (protective effect ?). No assessment expected soon because difficult to reach in winter. However, the site has already survived major earthquake: minor damage expected
Arslantepe: major damage to the nearby city of Malatya. The site seem quite well protected: minor damage expected.
Great Mosque and Hospital of Divriği: seems to be further afar from eartquake: no information on damage in the area.

And in Syria (sites that I have not visited)
Aleppo: major damage already reported in the news.
Ancient villages of northern Syria: no information found but seem to be fragile: major damage expected, if not already destroyed by war...

Many TL sites could be affected:
- the underground structure of Gaziantep (major damage in the city, including near destruction of Gaziantep castle !)
- Zeugma (damages depend on whether the protective roof stand)
- Sanliurfa (major damage in the city, no news of the TL area)
- Yesemek quarry (would probably not be affected to much by earthquake, as most sculpture are already on the ground)
- I have not visited other TL sites in the area which could have been damaged...

To anyone who was planning a visit to the region: postpone it at least for a year or even more. The life cannot be return to normal there in less than a full year. But after a year or so, please visit the area and support the local to help them economically.

First report from the ground about the ancient sites: https://www.al-monitor.com/originals/2023/02/prehistoric-world-heritage-site-gobekli-tepe-survives-turkey-earthquakes


No damage


Some damage

Massive earthquakes damage historic buildings in Turkey, Syria

Non WHS cultural sites that have been damaged:
- Gaziantep Castle
- The 17th century Şirvani Mosque in Gaziantep
- Grand Mosques of Kahramanmaraş, Adıyaman and Şanlıurfa
- The dome of the 17th-century Yeni Mosque in the city of Malatya
- St. Paul Orthodox Church in Antakya & Antakya Synagogue
- Cathedral of the Annunciation in İskenderun

Announcement from Ministry of Culture:

- Gaziantep, Sanliurfa, Kilis, Osmaniye, Diyarbakir, Adana - no damage in museum buildings and collections
- Malatya, Adiyaman, Kahramanmaras, Elbistan - minor damage in museum buildings and no damage in collections
- Archeology Museum in Hatay - moderate damage
- Arslantepe - minor damage
- Nemrut Dag - no damage
- Gobeklitepe - no damage
- Diyarbakir Castle and St. George Church - minor/moderate damage
- Gaziantep Castle - moderate/severe damage
- Malatya New Mosque - severe damage
- Hatay State National Assembly - collapsed
- Habib-i Neccar Mosque Hatay - collapsed
- Adiyaman Syrian Church - minor/ moderate damage
- Sanliurfa Balikli Lake - color of water turned to brown, minor/ moderate damage in nearby mosque
- Hatay St. Paul Orthodox Church - collapsed
- Iskenderun Orthodox church - severe damage
- Iskenderun Italian Latin Catholic Church - collapsed
- Many historical houses in Gaziantep and Hatay collapsed

https://twitter.com/ahmetcile/status/1623750994126307330?s=46&t=VKCVQb7GOAHV_63lWRVhPA

An update on the damages, including several (T)WHS: https://www.theartnewspaper.com/2023/02/27/full-scale-of-damage-to-turkish-and-syrian-heritage-emerges-after-devastating-earthquakes