Bitly links never expire.
When you delete a link, it no longer works — anyone clicking on the link or scanning a QR Code associated with it will get a 404 error message. If you hide the link instead, the short link is hidden from view in the Bitly app but continues to work. Its engagement data remains in Bitly.
Links that use a custom domain continue to work as long as your DNS is still pointing at Bitly and the custom domain is attached to a Bitly account.