I have trees whose trunk is physically deformed by the vines...it will not come out by itself. You'll have to judge on your own.

Hmm, that's what I was afraid of. Might have to call the tree doctor. I'm still holding out hope that it's a native Honeysuckle so I can just leave it alone as nature intended. The bottoms of the leaves are not "whitish" like I've read, but green like the tops, which is why I think it's Japanese. But the leaves are new, so who knows. We'll see when it blooms.

I'm making the wild guess that you're an american (may not be true), in which case your native honeysuckles have roundish to ovalish leaves that are not hairy and which feel more like a holly leaf (when mature) than the more fibrous leaves of japanese honeysuckle. But yes, when it blooms it's pretty obvious.

Yeah, Tennessee, Zone 7a. Thanks for the ID tips!

If you have the time and energy, you can cut the vines, as close to the ground as comfortable to you, paint the stumps with liquid weed killer, and the tops will come down, eventually, or might be pulled down. I drilled holes in the stump of a trumpet vine and filled the holes with weed killer, so that should work with honeysuckle. Shame to do such to a native species, but Japanese Honeysuckle is a tremendous pest in my gardens.