Qatar Airways (QR) is the only airline based in the Middle East that uses the Airbus A319 on Commercial / Revenue Services.
They currently have two A319 LR aircraft, registered A7-CJA and CJB. They also have a VIP configured version of the same aircraft, used primarily by the Royal Family of Qatar, although it is "technically" avilable for lease to external parties.
As far as I am aware, the only other A319's in the Gulf are corporate aircraft.
(In carrying out this assessment, I have considered EK, SV, KU, MS, IR, GF, EY, QR, RB, WY, RJ, ME. - I have probably missed out someone, or gotten one wrong).
In North Africa, TunisAir and its subsidiary, TunInter, also operate a couple of A319's.