No starter checks do not expire as long as you have an active checking account for those starter checks.
Starter checks are meant to be used until you can order regular checks or have your regular checks from the bank arrive in the mail.
But you can use the starter checks to pay bills, pay people, pay rent etc as long as you have an active checking account for the starter checks and as long as the money is in the bank.
As soon as you can order some regular checks though because some places and businesses may be reluctant to take starter checks without your name and address on them.
However most places and business will take starter checks.