If its asking you to verify your payment method, it in no way implies you will be charged for anything, just that a verified payment method is required.
You would only be made to pay if there is a pending charge on your account that needs to be completed.
Otherwise, a verified payment method is required to make use of the iTunes and App store services which includes downloading free apps and updating apps, but does not in any way mean you will need to pay for anything.
Once the payment method has been verified and thee are no pending charges or active subscriptions on your account, it can be removed and should not ask for it again unless there's a future issue with a charge or subscription.