Do you have to remove all low quality content?

[Henry Lawson]

It depends what you mean by low-quality content. If it's just poorly written and less interesting, but still has a healthy balance of keywords, you might benefit from hanging onto it (at least until you've got enough high-quality content to push it aside).

If your content is full of harmful backlinks, overly saturated with keywords, etc, then it's got to go! It hurts you page every second it's still up.