First of all, your iPhone must be jailbroken and running SSH. There is currently no built-in functionality in Shazam to export tagged songs or backup tagged songs to a web account.
SSH is free and can be found on the Cydia installer. Connect to your Wi-Fi network and look for the name of your network under Settings -> Wi-Fi. Select the blue arrow for more information about your network and make a note of the IP address of your iPhone.
Connect to your iPhone from a computer using an SSH client. The username is 'root' and the default password is 'alpine' to connect.