I haven't been able to find any studies on the effects of poor air quality on animals, but I was able to find quite a bit of general guidance. The American Veterinary Medical Association says this (and the guidance is very similar to other sources):
As irritating as smoke can be to people, it can cause health problems for animals as well. Smoke from wildfires and other large blazes affects pets, horses, livestock and wildlife. If you can see or feel the effects of smoke yourself, you also should take precautions to keep your animals – both pets and livestock – safe.
Animals with cardiovascular or respiratory disease are especially at risk from smoke and should be closely watched during all periods of poor air quality. Look for the following signs of possible smoke or dust irritation in animals. If any of your animals are experiencing any of these signs, please consult your veterinarian.
Coughing or gagging
Difficulty breathing, including open mouth breathing and increased noise when breathing
Eye irritation and excessive watering
Inflammation of throat or mouth
Increased breathing rate
Fatigue or weakness
Disorientation or stumbling
Reduced appetite and/or thirst
Tips to protect pets
Keep pets indoors as much as possible, and keep your windows shut.
Birds are particularly susceptible and should not be allowed outside when smoke or particulate matter are present.
Let dogs and cats outside only for brief bathroom breaks if air quality alerts are in effect.
Avoid intense outdoor exercise during periods of poor air quality. Exercise pets when dust and smoke has settled.
Have a pet evacuation kit ready, and include your animals in your disaster preparedness planning.
So essentially the guidance for pets is the same as for people, and you should avoid exposing them to outside air when the air quality is poor enough that you would avoid exposing yourself to it. For birds you may want to avoid exposing them even if you're comfortable with the current air quality since they are especially susceptible. Same goes for if you're healthy but your pet has risk factors.
The air is forecast to be in the "unhealthy for at-risk groups" level here tomorrow, which is a considerable improvement over the current state. Assuming that actually happens, I intend to go outside with my kitty for about 20 minutes. I'll watch her while outside and after we go in to see if she shows any discomfort. I also plan to avoid encouraging her to run around, since that would make her breathe faster and increase her exposure. If she tolerates that happily then maybe we'll repeat it throughout the day a few times.
Since she's healthy and not too young or old, I feel this is an acceptable trade-off between the health risks and alleviating her boredom. It would probably be fine to let her lounge outside longer than that, but I personally find it uncomfortable to breathe air outside of the good/green zone, so I'm limiting the time outside for my sake as much as hers.