You may need to save & backup your iPhone SMS on computer to make room for new SMS messages as iPhone has limit on number SMS massages can be stored or you simply want to achieve iPhone SMS messages for future references. Unfortunately there is no such feature in Apple iTunes for backing up iphone SMS messages.
However here is freeware third party app Syphone for Mac OS iPhone Users which lets you view, save, and back up your SMS messages with Syphone you can also archive and view SMS conversations, including those that have been deleted from the iPhone.
Syphone uses the familiar iChat format to display archived SMS messages. In addition, if you have multiple iPhones synced to your computer, Syphone allows you to view and archive messages from any of the iPhones.
Features of Syphone iPhone SMS Management Utility
1. View synced iPhone SMS messages on your Macintosh
2. Archive SMS messages
3. Import/export SMS messages
4. View SMS messages offline in an iChat-like interface with selectable colors
5. Open Address Book contacts
6. View the number of sent and received messages for each contact
7. Support multiple iPhones
Screenshot of Syphone iPhone SMS Management Utility