Checkout pages on Shopify are not editable unless you have the Shopify Plus plan. The Shopify Plus plan is $2000/month and up though!
So how can a regular Shopify owner remove the discount code field from Shopify checkout?
Simple – remove all discount codes and have no discounts in your store.
If there are any discounts that are active the field will show up:
If you have no discount codes, it will not show up: