Why is my mountaineer overheating and continue firing

Over heating can be due to a bad thermostat, a plugged radiator, a bad water pump. and a bad head gasket. The engine would continue to run, fire until it locked up if it overheated to the point of locking the engine up.

If it is running after you turn it off (dieseling" or "run-on). When you shut the ignition off a couple of the cylinders to keep igniting, which in turn keeps the engine running, although "sputtering" pretty badly. This can be caused by excessive carbon build-up in the combustion chambers or on the piston tops which when hot, acts like little glow plugs. This is common on engines that run way too rich or have oil burning problems. Dieseling is a common problem with high compression / high performance engines, especially in warm weather or when the engine is hot. In this case, it is usually caused by running "cheap" gas with inadequate octane. In this case it can usually be remedied by switching to a higher octane fuel. In your case its because its running hot. Need to fix the issue before you end up warping a head & blowing a head gasket, in-fact I would go ahead and perform a compression check on all cylinders and a exhaust in coolant test too make sure it has not yet of if it has already decide if you want to fix or buy another car as its costly to repair if a shop does it (labour wise).

What will cause the thermostat blow on my 2003 mercury mountaineer? Would that be cause the thermostat is not opening up all the way which is building up pressure