I spent a few weekends in gloves, long sleeves, pants, mask, and googles with a hacksaw. The vine was three vines twined around the tree trunk, each vine about the width of my lower arm. I cut out chunks of the vine at the base of the trunk, and it was immensely satisfying watching the vine's leaves droop and wither. A bit unnerving when some extra windy days scattered the dying leaves across the yard.
Anyway, now (well, not literally right now as it's winter) I just take the occasional morning or afternoon and pare back any new growths and dig up more of the buried vine roots. Not sure what, if anything, I can do about the dead vine still wrapped around the tree, but at least the tree isn't competing with the vine for sunlight and air anymore.