One of them has a fence around it to keep people out and a sign declaring it to be the property of the Hellenic Republic:
I can't find any information about this building on the Interwebs but I'd love to know its significance and history. Was it a church? An embassy? Why does Greece own it? Is it slated for restoration?
You can see its silhouette in the centre here, above the Old Town and between Narikala and Kartlis Deda, just to the right of a larger ruin which may or may not be related:
I would say this is a very prominent, and even prestigious location in Tbilisi.