Absorption refrigerators use heat to power their cooling cycle. If you remove the outside access panels while the refrigerator is operating are the cooling tubes on are the rear of the fridge warm or hot to the touch?
If the rear plumbing isn't hot, it's a control problem that's keeping the boiler's heat source off and the cause can be tracked down and fixed.
If the rear plumbing is hot, i.e. the boiler is working but the refrigerator isn't cooling, the cooling unit has failed. Make sure there's nothing obstructing the airflow around the cooling coils and through the roof vent so the excess heat can dissipate properly. If these are OK the only remaining solution is to replace the cooling unit (expensive) or the refrigerator.
While you're back there, check for an ammonia smell or yellow deposits around the pipes. These are additional signs of a failed cooling unit.
