It's not really clear when airspace restrictions will go into effect but based on how suddenly this was announced my educated guess would be sooner rather than later.
Qatar will be able to continue although its flight plan will be less than desirable.
In order to operate, Qatar Airways would have to fly over Iran, Iraq and Jordan for flights to the Middle East or over Iran and Turkey for flights destined for Europe....
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Above from
Saudi-Led Coalition Cuts Diplomatic Ties with Qatar, Shuts Down Access to Airspace
Four countries – Bahrain, Egypt, Saudi Arabia, and the United Arab Emirates – announced early Monday that they were breaking off diplomatic relations with Qatar over that country’s support for Islamist groups and its relationship with Iran.
The four nations also said that would cut off air and sea traffic to Qatar as well as refuse Qatari aircraft access to their airspace, a move that, among other things, would impede Qatar Airways’ ability to operate. The Saudi government also said it would shut down land crossings with its neighbor....
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