Never burn wood with poison vines on it, the oil can be inhaled as it burns and inhaled causing a whole new allergic issue as the allergen compounds in the urushiol oil get carried into the air and inhaled into your throat, mouth, lungs, etc.
It can also settle on unsuspecting bystanders as well.
Back about 30 years ago this area had a huge winter ice and wind storm, the aftermath left huge old growth trees down all over the area. The place I was working for had four huge trees come down across a row of vehicles parked by a fence. The vehicles weren't damaged much as the smaller branches supported much of the leaning trees but the trees had to come down. The far side of the fence was a land locked field that butted up to an abandoned rail yard. This was in mid December and there were no leaves on anything. The four trees were huge hickory and walnut, each one roughly 40" at the base. Three of the four were covered in huge fuzzy looking vines, which were likely what choked out the trees and left them compromised enough to topple in the wind and ice storm. The ice was all but off the downed trees by the time we got back there to clean up the mess.
We weren't thinking much about poison anything, it was cold, damp, and windy and we just wanted it done and over with.
We started by moving the trucks that we could right away, then started to take apart the trees from the top down. Once enough weight was off the tops of the trees they sort of up righted themselves back up on the other side of the fence, but not enough to clear the fence for repair. Four of us spent the next 15 hours cutting up all four trees. Never once did we think about any of the vines or any poison. Two days later one guy called out sick saying he's covered with some sort of rash on his arms, face, and legs. The next day the other two were complaining that they had gotten a rash on their arms, hands, face and anywhere they had touched with their hands that day. I didn't get any reaction for almost four days when I got a bad rash on my arms and between my fingers. It looked like burn blisters up and down the inside of both arms, a little spot on my neck, and between my fingers. It lasted for a week and after nothing seemed to help it, I took the advice of an old timer I knew that said to scrub it with rubbing alcohol till all the runny blisters were gone and let it dry. I did that and it was nearly completely gone in two days.
I was wearing a Carhartt insulated coveralls that day, I was covered with a full face ski mask, scarf, and gloves.
I think I was fine the first day because I didn't get anything inside my coveralls, but I wore them again a few days later and had bundled them up in a roll in the back of my truck when I was done that day. I think that's when I got it.
Some of the guys just tossed their clothes, one guy was wearing his work uniform and a company jacket, he said he got it all over again even after washing everything twice so he tossed it all.
I can't say for sure what the poison was but it was most likely a combination. The following summer that area was full of Poison sumac and poison oak was growing all along the fence line on both sides. The vines were likely poison ivy as it was growing up and down all the nearby power line poles as well.
These days, poison ivy doesn't bother me much, but I'm careful to look for it and wash it off if I do touch any of it.
Poison sumac is more common here, but although it tears up most folks it does very little if anything at all to me.
In recent years we've had a sudden resurgence of poison oak here, the stuff pops up every where. Last summer I removed two barrels of it from around my house and yard. It seems to grow in amongst the Virginia creeper vines that pop up in the bushes every year.
I was told by someone who researches native and non-native plants here and they told me that there's some who believe that not only are most poison plants more dangerous at certain times of the year, but how they affect us can also be affected by our personal health at that given time, along with what we eat. The way it was explained to me is that if you eat something that you may be borderline sensitive too, or if your immune system is busy dealing with another issue, your more likely to get a contact reaction to a poison plant. I'm no expert but it does seem to make a lot of sense.
I do know that getting a poison ivy rash on top of bad sunburn is particularly nasty, (I had an ex who burned easily, spent a day on the boat with me, then hopped off the boat on a small island to take a bathroom break, and basically sat in poison sumac, and somehow it was all my fault).
These days I don't much worry about the stuff but also don't rolling in it either. Don't touch it, if you do, wash your hands, clean your clothes of toss them if its bad enough, and if cutting wood, a face mask and full body protection is likely in order.
The sap and wet chips that come off those huge vines is bad, and likely far worse than what's on the leaves during the summer.
A buddy who works for the local municipality clearing power lines and fire breaks said they use something called Zanfel to wash clothes after working around poison vines.
A few guys use dish soap to wash their clothes, hands, and equipment with as well.
The main thing to remember is that urushiol oil is not a living thing, its an oil, you can kill the plant but not the oil.
It simply has to be removed or washed away in a timely manner to prevent a reaction.
If it remains on the skin or clothes you will likely get some degree of a reaction either now or later.
Even dead, dry vines contain the oil and can be almost just as bad as live vines.
Alcohol, many dish soaps, as well as Zanfel will remove the oil.
Use hot water to wash clothes and equipment, cold water to wash yourself with and wash anything you touched after contact with the plant or vines.