This article explains how to use the iPhone's Visual Voicemail feature to delete, undelete, and permanently remove your voicemails. The instructions in this article apply to iPhones running iOS 8 and later.
How do I delete voicemail on iPhone?
Tap the Phone app to launch it. If you're already in the app and have just listened to a voicemail that you now want to delete, skip to step 3.
Tap Voicemail in the lower right corner of the screen.
Find the voicemail you want to delete. Tap it once to reveal the options below. You can also swipe from right to left to reveal the Delete button.