Overnight a number of neighbors to Qatar have decided to cut their connections with the state.  Saudi Arabia, Bahrain, the UAE, and Egypt announced they were breaking diplomatic relations with Qatar and instructed their nationals to leave the emirate within 14 days.
Looking at the map one can see this is not some small spat. Qatar only has one friendly area and that is across the Gulf in Iran. This is awkward.
(source: FreeworldMaps.net)
Look at the airspace as of this writing (10:31 am EST). The area for air traffic is limited.
For a bit of perspective, here’s a picture with more context and regional scale. There’s a lot of air traffic using the Gulf.
A tiff that leads to closing of Saudi, Bahraini and UAE airspace poses serious limits on Qatar Airways’ operations. This means inbound and outbound flights for Qatar Airways to the north and northeast, must go out over the Gulf and, potentially, over Iran. This may work politically, but it means longer transit times.  Schedules will have to be tweaked.
Here is the in-flight Qatar fleet at this moment, between the Gulf and the EU. The flights across Saudi Arabia to Egypt will probably cease.
The issue in the region, as is frequently the case, comes down to Shia vs Sunni. Which means the aviation aspect simply gets lost in the grander difference of opinion.
The concern for the whole region must be that the issue escalates to include more countries. Especially those at the north end of the Gulf.