Once the poison ivy is dead it is still potent and able to cause a brisk skin rash. During removal you will still need to exercise caution with long sleeves, long pants, socks and gloves and avoid inadvertently brushing your face with gloves or plant etc. Immediately after dismembering the tree and disposing of the vines in a plastic sealed bag (unless you have a little used forest or ravine to walk them to) take a shower. If you suspect you've been brushed by it during removal be sure and wash area within 15 minutes and then resume your work. The easiest way to put the vines in a plastic bag is by putting your gloved hand inside the bag, then grabbing the vine with the bag and then inverting the bag over the vines.
The urushiol oil is stable for years so do not burn any wood that had vines directly on it or within a couple feet of it. Realize that die back of the vine may be misleading as to where it had been. Inhaling this smoke can put you in the hospital at worst and at least the smoke's contact with your skin can leave you with a more intense rash than touching it.
I work in the emergency department and the worst rash I ever saw was on a woman using this wood in her woodstove.
Hope this helps.