Burning is the least environmentally friendly option. Possibly the quickest way to kill them off is to let your sheep into the orchard. Most breeds of sheep love fruit trees, especially apples, and the whole tree is good eating for them, except the wood of the trunk. They would eat off the bark as high as they could reach, also low branches and leaves. You would still have to pull out the ones you don't want, and the sheep could then finish off eating the leaves and bark once they are felled. If you do stack them at any stage and burn them, please check carefully for nesting birds and hedgehogs - they can make their homes in a pile of dead branches really quickly.
I like the idea of offering 'dig your own' trees for free, but time is passing so advertise that soon.