Bitly links never expire.
If you hide links in your Bitly account, the short links will continue to work and their engagement data remains in Bitly. If you delete a link, it will no longer work — anyone clicking on the link or scanning a QR Code associated with that link will get a 404 error message.
If you use a custom domain to shorten your links, they'll continue to work as long as your DNS is still pointing at Bitly and the custom domain is attached to a Bitly account.