All else equal, do inactive credit card accounts with a zero balance tend to help or hurt the credit score. Thanks in advance for your answers!
They account for some age but, to a marginal extent just as "closed" accounts. Try using a few and then paying it off to invoke them to an open status (they may be closed so call first).
They are not hurting your FICO score; however, if the credit issuer closes the accounts due to inactivity, that will likely lower your credit utilization ratio and damage your score. Unfortunately, a lot of credit issuers are doing that right now. So, as apex suggested, try using the accounts from time to time for a purchase and then pay it off in full. You want to make sure you keep the history and available credit associated with the card.