When would the PBI's finish in the product backlog?

The Product Owner is responsible for the product backlog. The PBI's or otherwise known as the user stories, epics or features are the contents of the product backlog including the product vision and the product goal.

The product backlog exists as long as the product exists.

A good instance when a product backlog is regarded as finished is when a product gets phased out from production. We can think of a mobile app, website or a webapp that used to be a top competitor in business, finance or social network category but now no longer available or existent on the app download store.