You can delete any card on the iPhone app by tapping either Drafts or Post Box at the bottom of your screen, depending on whether or not you've already sent the card. Then find the card you want to remove and swipe left on it, then tap "Trash."
You can delete cards from your Postbox as long as you're viewing them in the "list" view. Tap on the three lines at the top left of your screen to open your account menu, then tap "Post Box."
Find your card in its respective tab (sent, received, drafts, etc.) and make sure you're viewing the List View (you can toggle between thumbnail and list view by tapping the buttons at the top right corner of the screen). Then swipe left on the card you want to remove and tap the red "Trash" button to delete it.