It's a story about a person who comes out of the closet, but the problem is that he lives in the middle east. And now he's getting pursued and strives to keep himself alive.

Don't start the logline with "It's a story about" because this is a waste of five words that you can use better. Every logline will summarize "a story".

"A person who comes out of the closet": we really need a little more about this character. How old is he? Why does he come out NOW? Is there an inciting incident, trigger for this decision?

"the problem is that he lives in the middle east": gay people don't just have a hard time in the Middle East, therefore you will need to navigate carefully not to use stereotypes.

"Now he's getting pursued": by whom?