Hi, 

This appears to be Japanese Honeysuckle, which is an invasive plant. It could not be showing flowers because of age or if it is shaded by overhead shrubs perhaps. You can try to hand pull/cut/ and dig as you much as you can to remove it. You may need to carefully spray or paint on fresh cuts, of a systemic herbicide to gain initial control. Something with either glyphosate or triclopyr is typically what we would recommend, but if you don't want to use chemicals, you will need to just repeatedly pull any vines you see emerge. Honeysuckle is easily spread from birds and wildlife eating the berries and distributing them, so if you can monitor where it is and just stay on top of removal, that is the best way to try to manage it. 

Let us know if you have further questions. 

Emily