Apple iPhones use a proprietary voice mail service called Visual Voicemail. One of its drawbacks is that the iPhone users are prohibited from forwarding voice mails from either Visual Voicemail or the native AT&T. There are workarounds to this limitation, including using apps to manage voice mail and to forward all voice mails to another voice mail provider, namely Google Voice.
Use an Apple approved application to manage your voice mail. Apple's iPhone app store has several apps that enhance the iPhone's native voice mail capabilities. Fusion Voicemail (available as a free, ad-supported application or a premium, no-ads version) and Voicemail Inbox are two of the more popular voice-mail manager applications that include a function to forward voice mail messages vial email.
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Use Google Voice to manage your voice mail messages. Though not a true method of forwarding voice mails on your iPhone, but using Google Voice to manage your voice mail messages will allow you to take advantage of Google Voice's voice-mail system including text previews of your email and the ability to forward voice messages.
Log in to Google Voice. Add your iPhone to your Google Voice Phone List. Navigate to Settings and click on the Voice tab. Click "Add another phone" and follow the instructions to add your iPhone to the phones associated with your Google Voice number.
Enter _004_1[your Google Voice phone number]# and press "Send." For instance, if your Google Voice phone number were 415-555-1212, you would enter * 004 *14155551212#. Your iPhone would display a screen confirming that your voice mail was being forwarded to Google Voice.
Manage your incoming voice mail on the Google Voice site. Google Voice will send a transcript of incoming voice mail to your Google email address. From Google Voice, you can listen to, download and forward your voice mail messages via email.
To stop forwarding your iPhone voice mail to Google Voice, enter ##004#.
Enabling Google Voice on your iPhone will disable your iPhone's Visual Voicemail feature. Any voicemails handled by the Google Voice mail system will not show up in your iPhone's Visual Voicemail even after you turn off Google Voicemail.
Jailbroken iPhones have access to non-Apple approved app stores that include Voicemail Forwarder, which is another voicemail manager that allows forwarding via email. Jailbreaking an iPhone is legal, but it does violate Apple's warranty.