| First, NEVER burn poison ivy, poison oak or poison sumac no matter how long ago it was cut.
Under dry, cool conditions, the allergenic oil urushiol (which is in the plant resin) is stable for possibly hundreds of years.
Out in the elements, of course, the leaves degrade and are composted, and seasonal weather acts on the plants, but the branches and stems can sit on the ground and not break down much for a number of years. I would not trust that the urushiol in the branches has degraded.
The urushiol is pulled back out of the leaves in the fall, so if they are handled they are not nearly as dangerous as spring or summer leaves, but they are not completely urushiol free.
The reason for not burning the plants is that urushiol is an oil that is not volatile (will not evaporate). In a fire, droplets rise into the air on ashes and soot and will fall from the air onto your body. Smoke breathed into the lungs is considered to be very dangerous.
My credentials: I wrote "The Poison Oak & Poison Ivy Survival Guide."