Will it kill an oak next that's growing couple inches from it?
It depends on chemistry used and as k bush said, how sloppy you are. Generally when I see injury to nearby vegetation, it is either 1) root exudates 2) too much applied to hack area and you get runoff in the next rain and roots take up a soil active herbicide like imazapyr or 3) you habe root grafting between same species. Root grafting really isn’t a concern across species.