Peanut Allergy is the most prevalent food allergy in the US, where as many as 1.5 million people suffer from the disease (American Academy of Allergy, Asthma & Immunology). Peanut Allergy is the most common cause of food related death (Asthma and Allergy Foundation of America).
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To be honest, this is a tough one. I'm inclined to say that it could in those with airborne sensitivity as there are those who have had reactions from being near nuts that were being roasted or cooked.
I know that burning poison ivy can cause severe reactions as the oils get into the air and into the lungs.
Regarding peanuts--I would say it depends upon the proximity of the allergic person to the smoke.