The chemical/plant oil that makes poison ivy so toxic is urushiol. Nasty stuff. It stays active for years, on tools, gloves, and in dried stems of the chopped plant. I once walked though poison ivy on a hike, and got a rash on my hands 2 weeks later when I tied my boots.
There is a story of a museum that had poison ivy samples in its collection, and when someone touched them 50 years later, it still caused a rash.
I have no idea if a hot compost pile would render the urushiol benign, but any stems that are still laying around uncomposted would certainly still have enough oil in them to cause a rash. Wash you tools and gloves with a simple hot water and soap solution, and that should take care of it, but as for stems/leave of the plant itself . . . the permaculture solution: sheep or goats.
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Not sure where you're located, but here in New England the poison ivy has no leaves yet--but the roots, runners, etc. still have that nasty oil. I find that spring clean-outs are the time to be most careful; the plants are hardest to identify when they are leafless. Wear gloves. Wash with lots of soap and hot water, and Tecnu soap if you can.
Poison ivy roots can definitely give you the rash even when they are dormant and have no leaves. I've very sensitive to it, but when I am exposed, I always wash with whatever soap is available and then don't get it. Special soap is not needed.
Once a friend and I sat in the winter in New York state making shapes out of nice long roots and runners that had no leaves. Then I remembered that in the summer that area had been full of poison ivy, so we went home and washed throughly with soap. I didn't get the rash that time at all, but the process of drawing his hands through his jacket sleeves left a residue, and a few days later my friend got such bad rashes on his arms that he had to go to the hospital.
Since it's a natural plant oil, I would think it composts fully and turns into harmless materials, but because I'm so sensitive to urushiol, if I had poison ivy vines to compost I'd put them in a separate brush pile in an out of the way place that I'd never have to come back to.
Since you say there were roots in the compost, you might have live poison ivy growing in your compost pile. I'd recommend waiting till it starts to become recognisable in spring, and then ripping out as much as you can, disposing of it where it won't get touched by anyone, and then washing your clothes and hands and body, paying special attention to be paranoid about anything that might have touched anything, like tools, doorknobs, shower curtains, and taps. Just soap and rinse anything that might be contaminated.
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