If a flight was, say, en route to Seoul, but could only make it to the closest available airstrip and that was Pyongyang, would that be allowed? What would they be met with on the ground?
I can see at least three questions here and only one of them is fully aviation-related. First, can an airliner legally divert to Pyongyang in an emergency, under the laws of its own country? For US airlines, the answer is yes. 14 CFR 91 SFAR 79 bans flights over North Korea, but does allow an exception in an emergency:
2. Flight Prohibition. Except as provided in paragraphs 3 and 4 of this SFAR, no person described in paragraph 1 may conduct flight operations through the Pyongyang FIR west of 132 degrees east longitude.
4. Emergency situations. In an emergency that requires immediate decision and action for the safety of the flight, the pilot in command on an aircraft may deviate from this SFAR to the extent required by that emergency.
Second, will North Korea allow the airliner to land? This is a political question, not an aviation one. Either they say yes or they say no. If they say no to an airliner in distress, how will they enforce that? Are they willing to shoot it down rather than let it land? Would that decision depend on the country of the airliner? Can they even make that decision within minutes, or is it already made and they simply initiate a defined procedure? The pilot is probably just going to try to land anyway; assuming the alternative is a crash, he would have little to lose.
Finally, what will the 'reception committee' on the ground look like? No idea, this is a political/military question.
But practically speaking, this situation should never arise anyway (ignoring movie plot stuff like hacked navigation systems). If your country isn't friendly or at least neutral to North Korea, your airlines simply won't plan flights where the only emergency landing possible is in Pyongyang, and for US carriers specifically it would be illegal anyway.
In an emergency that requires landing as soon as possible an aircraft can (and should) land at the nearest suitable airport, laws and international incidents be damned. If your 747 is on fire then being on the ground at an airport in North Korea is preferable to exploding in the air or winding up in the ocean or on the ground somewhere that's not an airport.
As Pondlife noted the regulations prohibiting flight over North Korea contain a specific provision lifting the restriction in the event of an emergency, in addition to FAR 91.3's provision that:
In an in-flight emergency requiring immediate action, the pilot in command may deviate from any rule of this part to the extent required to meet that emergency.
Everything else isn't an aviation problem: This is why countries have diplomats (and, if necessary, their armed forces), however as North Korea (or more properly "The Democratic People’s Republic of Korea") has submitted a "Notice of Adherence to the Chicago Convention" they're bound by its requirements, specifically Article 25 which states in part:
Each contracting State undertakes to provide such measures of assistance to aircraft in distress in its territory as it may find practicable, and to permit, subject to control by its own authorities, the owners of the aircraft or authorities of the State in which the aircraft is registered to provide such measures of assistance as may be necessitated by the circumstances.
There are very few circumstances where allowing an aircraft in distress to land would be "not practicable", though nothing requires them to let the passengers/crew off the aircraft, provide other support services, or make it "convenient" for the passengers & crew to get back home.