Image courtesy of iSpazio
While not available from the App Store, iPhone owners who have Jailbreaked their devices can enjoy Copy and Paste with the Clippy application from iSpazio.
Once installed, Clippy creates a new toolbar above the number keyboard in all applications. Swiping a finger over text will start to highlight it, after which one can copy-paste using the buttons in the new keyboard.
The extension is known to cause problems with Intelliscreen and iRealSMS.
Available through Cydia (which you can get by Jailbreaking) from the iSpazio repository.