|Like I said before, don't burn it. If you cut up the vine and it dies the oils can still remain active for a very long time (in dry climates, for years). Wear gloves and use a hacksaw to cut it up so that you are not breathing in any dust. Designate a brush pile that can sit for a while (months or a year) until the oil dissipates. If you are very allergic (like I am) you might want to hire it done.
I am not an expert but have talked to lots of friends and neighbors and done lots of reading since coming in contact with poison oak.