Can an airplane engine fire be extinguished without shutting down the engine?
In most modern jets (e.g. Boeing 757/767) when you pull the fire handle the engine shuts down (fuel, hydraulic, etc., are cutoff). But pulling the fire handle only "arms" the extinguishing system.  You have to rotate the handle (left or right) to discharge a bottle. (there are other bottle configurations based on the airplane type, but this is the general scheme)

Usually, after pulling the fire handle (before rotating it left or right) you wait momentarily to determine if the overheat or fire warning lights have extinguished.  If not, you rotate the handle to discharge the extinguisher.

So, if you are asking about modern jets, the answer would be no, you can't discharge the extinguishing agent without shutting down the engine first.

I am not aware of a modern design that would allow the engine to continue operating after activating the fire extinguishing system.