First I'll point you to the Control section as a starting point. Your question is very good as to what do you do once the plant is dead. And your fears are well-grounded as the vine does indeed contain urushiol oil (the toxic chemical which causes the rash) and must be handled with care.

I have found that RoundUp on a large area can take several applications. You'll need to be patient to ensure complete removal. A very important tip - don't use powered chopping tools (weed whacker, chain saw) on these plants as you'll send little bits of plant and urushiol oil into the air (not good!). Hand tools are best. Be sure to have a big bucket of soap and water available for cleanup plus the water hose. Cleanup requires lots of water. Consider getting a pair of heavy-duty to the elbow vinyl gloves for working around the plants. Have plenty of trash bags on hand as you want to bag up what you are removing. Don't burn and don't compost.

For plants that are running up a tree, cut a 2 foot section out of the vine, pry it up, and throw it in the trash (bagged of course - don't want any surprises there). Let the plant die. If you wait a couple years, the dead vine will eventually start falling out. If it's an old hairy vine, its very difficult to pull off a tree so waiting is a better idea.

For plants on the ground, you'll want to get the vine up. Also, try to get as much of the root as possible. In an earlier post, I mentioned how long it took me to finally rid an azalea bush of a recurring poison ivy plant. I cut the root, pulled what i could, sprayed on Roundup, etc. It took a few years before I could declare total victory on that particular plant. I couldn't get all the root out without damaging the azalea which factored in on how long it took to eradicate. Persistence is important here.

That's about it for me. Anyone else?