My phone's screen died but it still works. How can I get the text messages from it?
AndroidLost is an app that might fit your need. Though the aim of the app is different and can do much more, the abilty of the app to read sms and send it to email is an interesting 
solution to your problem.

Read its FAQ and Getting Started sections - you need not even touch your phone as long as it is connected to Internet for installing this app.