Well, burning it is definitely dangerous. You can hear many anecdotal stories of how bad it is, or you can listen to the
CDC:
Burning these poisonous plants can be very dangerous because the allergens can be inhaled, causing lung irritation.
If the guy can push it all into an unused corner of his land and leave it as a brush pile, it will rot down in a year or two, especially if the climate is not too dry. And the urushiol (the irritant oil) will get composted and become harmless. He probably thinks a brush pile is unsightly, but I can't think of any other reason against them except they can take up a lot of space.
I'm very very allergic to poison ivy, but I do find that if I wash thoroughly with any kind of
soap or detergent after touching it, I won't get the rash. I just think rationally about what objects, shoes, clothes, etc, could have touched it and then wash everything that might be carrying urushiol. It's gotten so I never get the rash after I know I've been exposed, but sometimes when I don't (because eastern poison ivy is rampant in Cape Cod where I spend a few weeks a year). However my one
experience with poison oak in the west was more difficult to understand and control.