How can I remove my Payment from my iPhone

Do you have any subscriptions?

Apple ID without a credit card  

Apple ID - If you can‘t choose None when you edit your Apple ID payment information

From my knowledge, Apple requires you to have a credit card on file to make purchases, even if you make free downloads from iTunes and the App Store. Contact Apple Support to confirm this, but I am almost certain this is correct.

I have downloaded about 25 free apps (and only free ones) and have never had to enter any kind of payment method.