Provided that you are certain that the equipment ONLY cools the interior air and introduces no fresh air whatsoever, then it MIGHT be acceptable to leave it running, subject to it serving only a single room, or perhaps several rooms but with an inside coil and fan for each room.
If however a central fan serves several rooms, then the equipment should be turned off since the air circulation will move fallout from a less protected part of your home and into your fallout shelter/inner refuge.

And in any case, it is most unlikely that you will still have utility service. And if you have a generator, is it connected to a bulk diesel fuel tank ? and if not, can you fuel it without exposing yourself to fallout ?

A serious level of fallout over a significant area implies a very serious situation indeed, and in such circumstances diesel fuel might be far to valuable to expend thus.