I've been wondering for awhile to myself, how long can one stay supersonic? How long before one ran out of fuel? What would the issues of staying supersonic be (aside from control)?
Depends on how much fuel that they have on board. You're not gonna be able to hold wing tanks or a centerline tank on that jet and still be able to fly supersonic, so we will have to assume that we can only use internal fuel. In a Block 50/52 F-16C, the jet holds about 7200 pounds of internal fuel with the tanks topped off. In full afterburner, the jet's F100 or F110 engine is burning approximately 800 pounds of fuel per minute. This gives a maximum of approx. nine minutes of flight time in full afterburner. So you have at most nine minutes of supersonic dash available to you.
As mentioned above, while an F-16 is capable of a maximum speed of Mach 2, it can't do this unless it's in a clean configuration, that is, it does not have any external stores on the jet which create a lot of parasite drag and prevents supersonic flight with ordnance on board. Typical mission speeds are high subsonic (500-600 KTAS) for cruise and combat maneuvering. Usually afterburning is utilized for brief periods in order to quickly accelerate the jet, such as to shorten takeoff distances, or to quickly replenish energy which has been bled off during maneuvering. Supersonic dash can be used to egress a FLOT or airspace deep inside enemy territory once, say a bombing mission is complete and you have been informed by an E-3 that there are enemy fighters scrambling towards your position. You jettison the external stores, save maybe a pair of AIM-9s or AIM-120s on the wingtip rails, push open the throttle and run like hell - and hope it's enough that you can make it to a friendly airbase or a tanker in time!
Fuel is one limitation, structural limits are another one. Fighters with good all-around view usually feature rounded canopy designs. This may lead to a stagnation point to be located on the canopy structure. Stagnation points are characterized by zero flow velocity, the kinetic energy of the flow is converted into inner energy of the fluid. As result, the fluid temperature is drastically increased. Canopy material and Mach number determine the period this temperature load can be sustained.
As reference, check aircraft designs like SR-71, Mig-25, or Concorde whose mission is characterized by permanent supersonic operation. These designs often feature flat, angled cockpit windows to avoid stagnation points.