What happens to incoming flights when the airport shuts down?
If a "terrorist takeover" like this were to happen in the real world...

...and the tower still had communication capability:

The tower controllers would advise pilots of the situation, and likely instruct them to return to the Approach frequency to coordinate their next steps (probably a landing at an alternate airport).
Presumably the tower would still have telephone communication as well, and be able to let Approach know to expect the planes to come back because of the emergency, otherwise the pilots would certainly inform the approach controllers of what's going on when they switch back to that frequency.

...and the tower has no communication capability:

With no response from the tower the usual thing for pilots to do is return to the Approach frequency to find out what's going on. The pilots could elect to attempt a landing anyway, treating the airport as an untowered field (this somewhat famously happened at KDCA, for somewhat different reasons), or proceed to an alternate airport.
If the weather required an instrument approach and the ILS radio signals are not transmitting this would be something the pilots could detect pretty easily (similarly if they were turned off while flying the approach there would be indications in the cockpit that the signal was lost and the pilot would execute the missed-approach procedure), and they would proceed to an alternate airport, again coordinating with the approach controllers.

...in any case:

No matter what the situation the aircraft would divert to their alternate airports long before they were in danger of running out of fuel. They would notify ATC of the situation and proceed to an alternate airport. (If not cleared by ATC to proceed to an alternate at some point any sane pilot will declare an emergency for low fuel and proceed on their emergency authority to make a safe landing somewhere.)



So basically there's some "artistic liberties" taken in Die Hard 2 in the interest of making it a movie you'd want to sit through.
A movie where a bunch of terrorists take over an airport and a bunch of flights divert to alternate airports (mildly inconveniencing the passengers) while the National Guard surrounds the airport and deals with the terrorists wouldn't sell nearly as many tickets!