I am using Google Chrome 40.02 and Intellij IDEA 13.1 and JetBrains IDE Support Extension 2.0.7. Live Edit plugin doesn't work and after click on debug button on the Intellij IDEA the browser open and notify me "JetBrains IDE Support" is debugging this tab but after one second this notification closed.
Please see https://intellij-support.jetbrains.com/entries/101809103-JavaScript-Debugging-Live-Edit-Chrome-extension-disconnects-in-a-few-seconds. You need to either upgrade Idea to v.14 (actually even IDEA 13.1.5 should include the fix) or downgrade Chrome to v. 36