Before we proceed with the means to fix your voicemail not working on iPhone, it’s of utmost importance to know if voicemail is set up in the first place.
Voicemail not deleting is another common issue with iPhone users. Again, it isn’t clear what causes this problem, but we do know several different methods to potentially fix it. If you cannot delete your voicemail on your iPhone, try these fixes.

To use a different greeting you already recorded or switch back to the default greeting: On your Android device, open the Voice app . At the top left, tap Menu Settings. In the Voicemail section, tap Voicemail greeting. Next to the greeting you want to use, tap More Set as active.
After setting the number on your voicemail, you must also record a personal voicemail greeting. Follow the steps below. Start the Phone app. if you don’t see the numeric keypad, tap “Keypad” at the bottom of the screen. Tap and hold the “1” for about two seconds until the phone automatically calls voicemail. Immediately tap “3” to access voicemail settings. Listen to the options. On most voicemail systems, tap “2” to start the process to record a new greeting. The current personal greeting will play; then tap “2” again to record a new greeting. Say the greeting you want, and tap “#” when you’re done. To save the greeting, tap “1.” The personal voicemail greeting has been set. Hangup the phone. How to setup voicemail using a Visual Voicemail Applications