Kill the ivy by cutting it at its base. Pull as much of the cut ivy as you can out of the tree. At that point you should dig up the roots.

Cut Ivy off high so leave stump tha tie chain around use tractor pull it up from ground.
Success pecan is great tree to have but they get way to tall for good nut production so shorten tree by one half it never grow that tall again and produce pecans more often. Cut shorter if can't get that high in tree I use 16 foot latter on some mature tall trees.
Now good time to do cutting because dormant.
Because Pecan tree so slow you be rewarded within next 5 years with heavy crop nuts and good shade tree nuts forum on water suckers as they age 3 to 5 years.

When I moved into our current house nearly every tree was like that. I cut off the ivy at the base. It took five years for all the vines to fall down, at some point you can start pulling them out. The longer you wait the easier it is.

Scott