I do believe that like some humans, some cats are allergic to bee stings.
An allergic cat 'might' have a bad allergic reaction to a bee sting and in the worst of cases that reaction might involve respiratory distress possibly leading to death if left untreated.
On the other hand, if the cat is not allergic to bee stings but gets enough of them in the wrong part of the body - neck mouth....that too might be dangerous as the neck can swell and hamper breathing.
(I guess the key question is, is your cat allergic to bees...and I'm assuming only an allergy test will reveal this).
If I were you....I'd get that screen anyway. That's 100% sure.
This no screen policy is beyond stupid...and is discriminatory on multiple levels. (Is this a city rule - which city I gotta know...or a condo/townhouse rule?)
(Richer people can afford air conditioning more easily....and I guess it's okay to have 6 hideous air conditioners popping out of a window but no screen door?) Beyond stupid, says Marko.
here's a link with pics that might be helpful.