Ok, it sounds like you may have a problematic installation. Sometimes there are circumstances where the sun shines on the side of the building and builds up a lot of heat in the organ chambers, or some other kind of unusual circumstance. I would guess that window fans might help in a situation like that. However, I would ask your organist about the openings that you would like to be closed, as they may be openings into swell chambers, which do need to be open for circulation.
If your instrument is as I am visualizing it, the screens (are they behind a set of facade, or "show" pipes? Or are the pipes completely hidden?) are a decorative part of the organ facade design. Organ pipes are not normally completely enclosed in airtight boxes except in rare instances. Normally the boxes which enclose pipes have venetian shutters on the front which allow them to be opened or closed by the organist in order to regulate the volume of sound speaking into the room. These are the shutters that I'm referring to that should be left open when the organ is not being played.
Normally the air flow through an instrument is as it is. I'm not an organ builder or an organ technician really other than the bits and pieces of information I've learned from organ builder friends, but I'll say again I've not heard of an organ with window fans in it, though that doesn't mean it never happens. It would seem to me that even if it's very hot in the organ chambers, as long as this heat is consistent, the tuning would not be a problem. I've been inside organ chambers that were stifling as the pipes were tuned. As long as the temperature returned to that stifling heat in the chambers, the tuning was ok. However, if there's a problem with the sun at certain times of the day or something else that makes the temperature wildly inconsistent, then it would be a problem. Consistency is the most important thing about temperature and tuning.
Does this help some? Perhaps a sketch or picture would be helpful to see if your situation is unusual.