What is the reason for the detour flight route from AUH to BEY?
It's for political reasons.The flight is clearly avoiding Syria and Israel. To fly a direct route between the two you must overfly either Syria or Israel. 

Looking at the flight paths from the last week or so of the two airlines that service that route and it becomes very clear. The flight in question is (as it would appear from flight paths) Etihad Airways, the flag carrier for Abu Dhabi. All flights between the two from Etihad take this same circuitous route. The reason is that they can't fly over Syria or Israel. The UAE (of which Abu Dhabi is a part of) does not recognize Israel as a state. Israeli citizens are not even allowed in the UAE. Etihad does not serve Israel nor are Israelis allowed to fly on it. Many Arab states do not allow El Al to overly them so it's a safe bet Etihad is not allowed to overly Israel's territory. 

In the past that would just mean flying over Syria instead. But Abu Dhabi is a member of the Arab League which has invoked sanctions against Syria. This would make it risky to fly over Syria and is most likely prohibited by the sanctions. So their only option is to go the long way around.

On the other hand, the Lebanese carrier Middle East Airlines/Air Liban also serves this route. Although Lebanon is a member of the Arab League, Lebanon refused to be party to the sanctions on Syria. Due to this they are still able to fly over Syrian airspace, so they have a much more direct route.



And it would appear that Etihad is still avoiding flying over northern Sinai since the Metrojet 9268 crash.