I don't think Scrum advice anything on it, this is a typical resource management issue. There are multiple options - your best bet would be to have a team member (BA) do the PO job till the organization find out a new PO for your team. Someone who owns the product or whose money is getting invested into the product should be the PO.
There's no specific advice, but by definition the Product Owner owns the product. This means that if the PO leaves an organization the product can, in theory, be expected to make the journey too. Of course this depends upon whether or not the PO is an entrepreneur, and not just a hired representative of stakeholder interests.