DON'T burn it- it's done awful things to people that way. And animals, I imagine.
Bag it up and have it hauled away with trash.
I'm not allergic, but I know that's just a statement about sensitivity- the oil is actually a toxin for everything alive, it's just that some are more sensitive than others.
One way to get rid of smaller plants is to wear good rubber gloves and use plastic bags to pull all of the plants you can up by the roots. Turn the bags inside out when you have pulled them up, and knot each bag, then toss into another bag to dispose.
I'm not fond of using all of this plastic, but it's a safe way for someone who is sensitive.
Remaining roots can be sprayed with ammonia or vinegar.
Another way- for the tree or vine form- is to cut through it as close to the ground as possible and soak it heavily in ammonia or vinegar- supposedly it pulls some of it in and will then weaken or die.
This has worked well for me when I've helped others out- I'm 'not allergic' and have a friend who is plagued by it and can't even touch English Ivy for her sensitivity to ivies.