In order to use Windows Live ID Delegated Authentication (which is the authentication mechanism used by the Messenger Web Toolkit) in Azure, you have to setup an external domain to be a CNAME pointer to yourapp.cloudapp.net. You then have to use the external domain for the Domain and Return URL fields in the Live Services Developer Portal. Also don't forget to use that external domain in your code as well. As a consequence however, your website has to be accessed via that external domain.
You can find a more detailed explanation and an example here (the article actually describes it for the Live Framework, but it goes the same for the Messenger Web Toolkit).