Thank you. But how do you make it work for video? Furthermore, is a video alert on the iPhone possible with a web based app? The project above is convulted in that you get a separate text message and email. I'm just looking for a simple live video feed of the door when someone presses your doorbell.
The e-mail is only a convenience, you wouldn't have to use is. The linked tutorial does not use text messages, it is actually pushing a notification to the phone. If a web service (like pushingBox) doesn't quite fill your video needs, you can certainly still make this work.
There's a number of ways in which you could implement this, from building an app ground up for iPhone and pushing your own notifications, to simply having a push service send you a link to a webpage with your web cam embedded.
The Arduino isn't going to be handling any kind of video stream, this would be done with another computer (or perhaps with a webcam that hooks right into your network? Do those exist?). The computer would need to run some type of video streaming service that would enable you to embed the video stream into a webpage. The linked tutorial seems to suggest that some webcams include software to make your webcam feed publicly accessible.
Everything else about the previously linked tutorial would work fine as far as using a push service and an Ethernet shield. A simple link in the notification to your web page with the video stream embedded would probably be sufficient for what you're wanting.
As far as whether you can have a "video alert" on an iPhone... my bet is no, as I haven't ever seen one. That's a result of Apple's implementation of their push notification standards. Implementing this idea as an Android widget might be a cool idea though...