It would take a lot more than the vines growing up the trunk of that tree. I do not think they would grow far enough up to strangle it out. But then if needed you could prune or just pull them off the tree and let it vine away from the tree.

No it wouldn't damage your tree, but the jasmine is rather annoying as a ground cover. We have it planted around our house and it just loves growing up into the japanese boxwoods and spreads all throughout our plant beds. It's such a pain, I wish we had never planted it! Plus it never blooms much on the ground much for some reason.

That tree looks like it could handle just about anything. I've never seen jasmine as a ground cover but I have one of those trumpet vines (the good kind, not the invasive kind, but I can't remember the name. Darn Alzheimer's!) that seems to prefer to be a ground cover. I have it on a fence but it doesn't seem to want to grow on the fence unless I tape (yes, tape) it to the fence. Might try that. Might be too scraggly though. Just a thought.

Carolina Jessamine does better in a sunny spot supported by a fence or something. Not sure how it would like the shade under a tree. I like the plant myself...I just wish that it could have a longer bloom time, but early spring will have to do.