In Poison Ivy and Poison Oak: Identification, Irradication and Home Remedies, Sandra Dark suggests killing poison ivy by hoeing or mowing it. The article also includes several tips for managing an outbreak if you get poison ivy on your skin. But the large poison ivy vine you have on your tree requires a more intensive approach.
Be sure to wear appropriate protective gear: rubber boots and gloves, a hat, goggles or faceshield and long pants. Start by clipping the stem of the vine with a long-handled pruning shears. Then, using a long-handled tool (maybe a channel-lock wrench), pull the stump if you can. If not, hoe off any new shoots as they grow.
The vine in the tree will die after the main stem has been cut. Rake up the leaves after they fall, being careful to wear the same protective gear. Do not burn the leaves or stems, as the smoke can cause severe allergic reactions when inhaled.
In winter, trim as much of the dead vines as you care to, remembering that touching even these dead vines could cause a reaction.
— Troy Griepentrog, senior associate editor