You don't have to have a payment method on file for downloading free content or updates, but if you do have one it must be valid for your region, have funds, and be able to accept test charges from the iTunes Store.
There are normally some hoops you can jump through as you create your iTunes account that workaround the need for a payment card or bank account, otherwise once the account is set up the iTunes Store may still expect you to provide one, even if you are obtaining free content or updates, which you can later remove. If you don't have or want to use a credit/debit card at this time you can try this method using iTunes on a computer...
- Sign out of the iTunes Store from the Store/Account item on the menu bar
- Go to Music > iTunes Store home page
- Go to Quick Links > Redeem
- Sign in with your Apple ID
- Go through the screens, accepting the EULA, etc. You should arrive at a Review Payment screen where one of the options is None - select this and continue. You shouldn't actually need to redeem a new voucher/card.
It is worth noting that one of the functions of the registration is to prove you have a valid billing address in the region whose store you are trying to use. You might also have difficulty selecting None as an option if you have any active subscriptions (see View, change, or cancel your subscriptions - Apple Support), an outstanding debit balance to the iTunes Store, there are unreleased authorization holds on the card (see About payment card authorization holds in the iTunes Store - Apple Support) or you are trying to switch to one regional store while being in a different region. See also If you can't choose None when you edit your Apple ID payment information - Apple Support and Payment methods that you can use in the iTunes Store, App Store, and iBooks Store - Apple Support.