The Discount Code box only appears in the cart when the buyer's order is actually eligible for some discount(s) you have created.
Our new cart (currently concluding beta testing) cannot be customized, but if you're using our classic cart, you can use the cart customization approach above to hide the Discount Code field and label.
Note that classic cart customization code is page-specific, so you'd need to add the customization code on every page where you have cart buttons, and it only works when the cart displays as an overlay within your page, so it would have no effect when the cart displays in "fallback" mode using a separate window/tab, such as when the buyer is using a mobile device or has Javascript disabled in their browser.
This help page explains how to create a purchase link that includes a discount code: