Careful!
Be very careful with Poison Ivy. I am a botanist and I also spend part of my time doing weed control for a state Department of Transportation. The Urushiol (the oil that causes the rash) in Poison Ivy is a very stable molecule and is active for years, even after the plant is dead and dry. And the reaction that you can get from the stems and especially the roots can be really severe. Perennial plants store nutrients and other water soluble molecules in their roots for the winter, so the concentration of Urushiol is very high in the roots when the plant is dormant and can really cause a terrible rash.
If I was faced with the problem that you have, I would probably spray the plant with Brush B Gone and let it die. Then very carefully (while wearing protective clothing of course) I would cut the stem near the ground and pull the vines down one at a time. I would dispose of them somewhere where they could naturally decompose over time and not cause further problems for anyone. And have a shower right after you handle the plants! Then I would cut down the tree. Oh, and as others have said, never, never burn it or you could inhale the oils and cause a rash in your lungs. Very serious.
I have also found that there are two very good products for dealing with the oils on your skin. One is called Tecnu and is about $8.00-$10.00 (US) for a small bottle and the other is called Zanfel, about $35.00 an ounce. Zanfel works great
after you get the rash by removing the oils from your skin, but can also work before as well. Zanfel is expensive, but when you are suffering from the rash there is no amount of money that is too great to get rid of it. I am told that Walgreens has a product similar to Zanfel for much less money, but I haven't seen it or tried it.
Good Luck and stay away from the P.I.!!!!!!!!!
For more information try this link with tons of great info:
http://poisonivy.aesir.com/