Have a look here:
Information on Conflict Zones | EASA
Generally, flights do not transit anywhere near Syria nor do they fly over airspace in the Baghdad FIR.
EASA/UKCAA/FAA continuously provide risk analysis of the situation. Any changes in risk result in the issuance of a NOTAM. (Notice to Airman)
Most flights will transit Turkish airspace and then into Iran then rejoin routing southbound into Kuwait and the Gulf. The opposite is true for most northbound/European bound flights.
Nothing is assured as this whole region is 'fluid' at the moment.