Yes, it is possible to have inhaled poison ivy when it is burned. Urushiol, the rash-causing oil, can be carried in the smoke from burning leaves or brush. For this reason, it is illegal to burn poison ivy, oak or sumac in several states.
It is impossible to tell if a poison ivy plant is wrapped around a tree once the leaves fall off. If you have firewood that has vines wrapped around them, and you are not sure if it could be poison ivy or oak, avoid using that firewood. If you think you have touched the potential poison ivy or oak vines, even if they are dead, you can wash with Tecnu Original Outdoor Cleanser or Tecnu Extreme to remove the rash-causing oil.
If you find yourself around burning poison plants, avoid breathing the smoke. If you think you have inhaled the oils, you may have poison ivy in your lungs. Symptoms may include difficulty breathing, a rash inside your mouth or lips, rash appearing in various spots of your body, and fever. You will want to seek medical help immediately as this can be a very serious condition.
Even if you haven't burned it, you can still get poison ivy or oak rash from firewood, see how here.