I'd be stunned if any of the infrastructure at Beruit airport is still working - the runways should be fine/ok, but the radars, the communications, the fuel distribution systems etc.. are probably fucked.
The big military airlift aircraft will be able to get in and try to rebuild capacity like air transportable radars and ATC systems, but it will take a bit of time. The fuel issue shouldn't be the problem it might otherwise be because there are lots of big international airports not far away that aircraft can pop into to fill up the tanks, but it's not going to help.
Sealift, which can bring in much larger quantities of aid - water purification systems, field hospitals, mobile bakeries, huge numbers of tents and sanitation systems - is a lot easier when you have a working port, and Beruit doesn't. Amphibious assault and logistics support ships can put stuff on the beach, but you need lots of lorries to move it to where it need to be.