Just tried this on my HTC One X. It does exactly what I was looking for, and works mostly fine. However it only turns the screen on for a split second when I tap the notification, and the notification disappears so I have to press the power button twice, after which the screen stays on. It's not a big deal though, as I'll only be using this occasionally when I want to play videos on the TV or similar.
This app would be even better if it could detect when HDMI was plugged in, and prevent HDMI output from stopping when the power button is pressed. That way I wouldn't have to worry about remembering to use the app instead of turning off the screen normally when I'm using HDMI output. Assuming you can detect when HDMI is plugged in, this should be doable with a wake lock.