No warrants do not have an expiration date and the police will still have your warrant in their database for many years.
Even though the police may stop trying to look for you for a simple warrant after say 4 to 5 years they can still arrest you for that warrant when they pull you over and run your drivers license or ID.
When they run your drivers license or ID in their computer system the warrant will still be there in the database and you'll be arrested even if it has been 30 to 50 years past the date the warrant was issued.
Warrants basically stay in the police database system for the rest of your life.
So once you die the warrant is then expired but so are you.
But as long as you're alive the warrant is still very active until you're arrested.