If my memory serves me correctly, then if you force push or update the branch in question in any way, GitHub will automatically update the pull request. If doing the force push would result in the pull request not being possible, then GitHub will tell you this.
You do not need to worry about updating the pull request as GitHub will take care of this for you.
This being said, doing a git push --force on any remote branch can cause mayhem for your coworkers who are also currently working on this branch. So you are correct to be shying away from doing a force push, but fortunately a GitHub pull request is not your biggest problem.