Many providers of shortened URLs claim that they will "never expire" (there is always the implied small print: so long as we do not decide to discontinue this service—there is no contract to be breached by a free service, regardless of "promises"—and remain in business).
A permanent URL is not necessarily a good thing. There are security implications, and obsolete short URLs remain in existence and may be circulated long after they cease to point to a relevant or even extant destination. Sometimes a short URL is useful simply to give someone over a telephone conversation for a one-off access or file download, and no longer needed within a couple of minutes.
Some URL shorteners offer a time-limited service, which will expire after a specified period. Services available include an ordinary, easy-to-say word as the URL with a lifetime from 5 minutes up to 24 hours, creation of a URL which will expire on a specified date or after a specified period, creation of a very-short-lived URL of only 5 characters for typing into a smartphone, restriction by the creator of the total number of uses of the URL, and password protection. A Microsoft Security Brief recommends the creation of short-lived URLs, but for reasons explicitly of security rather than convenience.