Answer
IT usually owns the technical accuracy, but business leadership has to own the recovery priorities and approval thresholds the runbook is built around. The strongest runbooks are shared assets, with named technical owners, business stakeholders, and a review cadence tied to infrastructure, application, and staffing change.
If nobody owns updates after the initial project, the runbook starts aging immediately. Buyers should treat documentation discipline as part of the software fit, not a separate afterthought.