I back the chipper up to it. Hook a sling to the poison ivy, attach that to the chipper. Get in the truck, pull forward 4 or 5 meters. Back up 4 or 5 meters. Get out, start the chipper. feed the ivy in. Be careful and all, face mask good idea, respirator, tyvec suit if you're allergic as most people are (I'm lucky and unaffected by the stuff).
The chipper will pull the vine out of the tree and it magically disappears. I charge a lot of money for this service and it usually takes just minutes, making it one of the most profitable things I do so I don't mind performing the service. Who else is gonna do it? Pretty tough to find ANYONE to deal with this particular situation so I more or less sell 100% of those jobs for whatever price I want to charge. $250 an hour is the rate I shoot for. It's a 15 minute thing, though.
It really helps if the ivy has been dead for awhile, like Dean's. I find the bigger the diameter of the stalk, the easier it is to get of the tree (no match against the puull of a truck). Some parts of the plant might be left in the tree, but that's just how that's gonna be. If the ivy had been dead awhile it comes off easier, too, but is dustier, though toxicity goes down over time. In this case you may want to pull the ivy completely off the tree, cut it up into regular lengths with long-handled lopping shears (no sawdust). Place lengths on the oldest, most beat-up tarp you have, roll them up, dumpsterize the whole thing, along with the long-sleeve shirt you were wearing, the gloves, and the particle face mask. Dispose of all of it. That's pretty fast, too.