How can I download WhatsApp without a payment on iPhone?

You do not get charged to download WhatsApp, but if you have a previous purchase that has not been paid, then obviously you will need to take care of that before you would be allowed to download anything else in the App Store whether it is free or not. Take not of step 2 here and Step 3 if there was a problem with a previous purchase.

If you can’t download or update apps on your iPhone or iPad - Apple Support

You cannot. A payment method is required for the download of any app, including WhatsApp.