I am unaware of any way to do this directly from the iPhone. There are 3rd party software programs that can access the voicemail and download it to the computer. A program that I have used in the past is TouchCopy. It works quite well for that, as well as accessing SMS messages.
This site contains user submitted content, comments and opinions and is for informational purposes only.
Apple disclaims any and all liability for the acts, omissions and conduct of any third parties in connection with or related to your use of the site.