This use means to harass someone or give them a hard time. So in this text the person doesn't want to 'yank the other around' so doesn't want to give them a hard time. Yank means to pull, usually aggressively and quickly, so gives the impression that a person is being treated badly and/or forced to do something they don't want to. Pulling or yanking them away from what they are comfortable with.

* yank someone or something around = to pull or jerk someone or something around. e.g. Don't yank Billy around so. You'll hurt him! Please don't yank the chairs around. Move them carefully. * yank someone around (Slang) = to harass someone; to give someone a hard time. e.g. Listen, I don't mean to yank you around all the time, but we have to have the drawings by Monday. Please stop yanking me around. because yanking you around or causing you pain is the absolute last thing = Because harrassing you or causing you pain is the last thing on my mind (i.e. something I will never do) :)