You can burn it and the poison ivy will be gone ….BUT…. Do Not let the smoke get near you, the oils or whatever it is that breaks you out travels in the smoke. I learned this the hard way when my 2 oldest sons burned alot of wood w/ ivy. Took prednisone shots to rid them of it because it got everywhere on their bodies.
If I were you, I’d simply take precautions. I, too, am extremely allergic to the oils and go systemic requiring shots and pills for weeks afterwards. When I found out about it, we had been cutting up a fallen tree that we drug through a poison ivy meadow. It can also soak through clothing if you have enough of it. I discovered this because my DH gave me the job of working loose the chain saw he’d gotten stuck in the trunk. I sat, straddled on the trunk, for a half hour wrestling this thing. Let’s just say, I was extremely grateful I was wearing a panty shield that day! But the rest of me was not so lucky.
I would use the wood, but coat my skin with soap. As a matter of fact, firemen use a product before going into forests to protect themselves, you could probably find this by doing a search. Jewelweed is also a natural neutralizer of poison ivy oil, you can find soap bars from natural suppliers. Then I would cover myself well, including gloves, and work in blocks of time where you will be able to finish working, strip outside (behind a curtain if you live close in), and go shower immediately. You can pick up your clothes with salad tongs and drop them into your washing machine carefully.
Be sure to wash off with soap anything your clothes brush against on the trip inside. Don’t forget doorknobs and shower handles may have something on them to wash off unless you just leave them open for yourselves or have someone else open them for you. If you even think you have gotten any stuff on you, put on some Ivarest, which works the best for me so far, or some benedryl type cream on the area right after contact and showering as well.
There is also a new product on the market. I saw it at Rite Aid and it neutralizes all the components of Poison Ivy and oak. It is suppose to also be a preventitive, so you can slather it on and work in exposed areas with out contracting it. I have to say „suppose to” because I am one of those people who are immune to Poison Ivy and Oak. But my wife isn’t and so I know it worked for her.