A business owner or someone explicitly given Activate business features permission can switch Instagram publishing off in Admin > Settings > Messages > Marketing. New publishing and scheduling stop. A scheduled item is checked again at its scheduled time. If publishing is off then, the item does not publish: it shows Didn't publish and its content goes back to a draft. Switching publishing on again does not bring it back, so schedule the draft again. To keep an item from failing, cancel or reschedule it before you switch publishing off. A request already accepted by Instagram may finish, so check the account if its result is uncertain.
Your drafts, publishing history and Instagram connection stay. You can still review results and use Cancel schedule with the appropriate staff access. Turning this setting off does not remove posts already on Instagram. The separate Marketing email switch stops new campaign sends, manual marketing emails and test emails; it does not change Instagram publishing.