This is a green topic, but let me be upfront-this is not my area of expertise. Like the great majority of people who work with wood, I've never had any reaction at all. Allergies are scary because there is more unknown than known, and every person will react differently-and potentially will have different reactions to the same conditions over time. Peanuts will kill some people or make others sick, but the majority of us can eat them all day long without a problem. Reactions to wood are equally varied.
To be safe when working with wood, emphasize the following common-sense concepts:
1) Take Reasonable Caution: Work with wood with reasonable and common-sense care-put on masks against dust, use gloves if you have sensitive skin, etc.
2) Stay Clean: Keep jobsites/workshops and yourself clean-don't let dust build up in your surroundings or on you, remove splinters promptly, etc. Be sure your shop or workspace is well-ventilated and that the fans aren't blowing dust onto you or others.
3) Be Aware: Monitor yourself-if you have trouble breathing or begin to itch or turn red, or have any other unusual reaction, take action: Use more protective covering, get cleaned up, get medication, and, if necessary, go to a doctor.
What did Mother Nature put into the wood that can cause us problems? I'm told that if it's a skin reaction, it is usually from the resins-the natural oils present in the wood-although apparently just like poison ivy, some trees produce strong chemicals as part of their natural growth. For example, black walnut is an incredibly popular wood that humans rarely react to at all; however, you can't use the wood dust for horse bedding-it will make them sick or even kill them. And while you might enjoy the fruit of a black cherry tree, no one wants to nibble the leaves-they contain a precursor of cyanide.
That said, any wood dust, even if it is not particularly chemically toxic, can pose a health risk, just as any inhaled solid particle would. The State of California has (of course) declared wood dust a carcinogen and requires flooring to be labeled with a warning. I have received advice that Western red cedar in particular has an extremely small dust particle size, and is therefore able to penetrate deeply into the lungs and cause reactions in people who are already asthma-sensitive.
And there can be naturally introduced "poisons" like mold or bacteria or artificially introduced chemicals like treatments against blue stain or fumigation against insects. You can have 10 people handle the same piece of wood and have no reaction or several different reactions.
And, unfortunately, just because something doesn't cause a reaction the first time you touch it doesn't mean you'll never have an allergic reaction. Your sensitivities can increase with exposure, just as lactose intolerance can suddenly afflict a long time lover of ice cream. Allergy-prone people should be more cautious in the woods they choose, and everyone should their limit exposure to sawdust of a new wood the first few times they work with it. Wear a mask, wear your gloves, and be safe!