NEVER BURN POISON IVY. Breathing the smoke will injure your lungs. You shouldn't even expose your skin to the smoke.
To get rid of poison ivy, you must remove it completely, including the roots.
This is a photo of poison ivy in autumn.
USING HERBICIDES TO KILL POISON IVY:
There are several chemicals available that will kill poison ivy. Check with your local lawn and garden store to see what they recommend. Poison ivy is a woody stem plant and normal weed killer will only make the leaves drop off but not kill the stem and roots. Protecting your body from the plant, cut the stalk and apply herbicide to the stump.
If you use a ground sterilizer, note that it kills every plant it touches and may keep anything growing in that area for years.
NOTE: If you use herbicides, the ivy must still be removed because the urushiol on the dead leaves, branches and roots will remain potent for at least a year.
PULLING POISON IVY OUT BY THE ROOTS:
You can remove poison ivy easier if you wait until after a rain, or if you soak the area completely with a hose or sprinkler. Make sure the water penetrates the ground deeply enough to soak the roots.
Wear a long sleeve shirt or jacket, and pants and socks to completely cover your legs and ankles. Wear boots or high topped shoes. Also wear goggles and heavy gloves long enough to cover your wrists. You probably will want to apply some poison ivy blocking lotion to your face.
Get plenty of large, heavy plastic garbage bags and then some smaller plastic grocery bags to put over your gloved hands.
Put a plastic grocery bag over each hand and pull up a plant by the root. Then put the vine into a large plastic garbage bag. Repeat until you have all the ivy pulled out.
When finished, take the garbage bag to your trash can; don't use the vines for compost. Remove the plastic grocery bags from your hands and open the door to your house with your gloves if they were never exposed to the poison ivy. If you think they were, carefully remove the gloves without touching the outside and use your bare hands to open your door. Poison ivy can be spread by urushiol on door handles, light switches, tools, clothes, etc.
Once you're indoors and making sure you haven't brushed against anything, strip out of your clothes. Put everything straight into the washer and don't mix with other clothing. Wash the clothes in cold water and soap.
Then get into the shower and wash with cool water only - no soap and no washcloth. Water dissolves the poisonous oil; soap and cloth can spread it to other areas. After you've showered for several minutes with plain water, then shower using soap and shampoo.
The next day, go back outside to see if you missed pulling any poison ivy roots. If you see some close to the ground where they would be hard to pull up, smother the roots with heavy carpet or some other heavy material, leaving it there for a month or longer.
NOTE: Don't forget to use a water hose to clean your tools, too, since you don't want to be exposed to the oil again Urushiol can remain active for at least a year.
Photo of poison ivy leaves in fall