How to get visual voicemails off iPhone 4 and store on computer?

This is old..but try it.
http://i.tuaw.com/2011/09/16/iphone-101-recovering-deleted-voicemails/

Oh..nevermind..i misread your question.

Voicemail is a file. It stores on your phone hard drive. You prolly need root access to retrieve them.

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Phoneview application... Page 7.

http://www.vcparks.com/docs/Welcome_To_PhoneView.pdf

 

My visual voicemails all reappeared on my new phone when I set it up. Maybe because I restored from backup.

 

Use iExplorer. It can pull them from the phone or an iTunes backup. In the past my voicemails were re-downloaded from the carrier, but this time setting up the iPhone 6 nuked all of my voicemails.

 

That's two for iexplorer. May not be cheap, but it works.