Microsoft will soon be launching support for Linux VM's to be hosted in Azure. So you could conceivably build a Linux VM, install and configure your MPI app and push it up into Azure.
However, you should probably first consider what it is that you want to achieve. If your goal is to take advantage of some of Azure's features (e.g. multi-instance availability & scale, for example), then you'll need to augment your app with some additional code. Thanks to the open-source community and Microsoft's rapidly increasing support for open-source, you could build additional functionality in, for example:
- Node.js running on Linux or Windows
- Perl, Python, Ruby and/or PHP all running on Linux or Windows
- .NET code running in a Windows worker role or on Mono in your Linux VM
If you're most interested in taking advantage of your Azure service config, dynamically adding/removing worker roles, using azure storage, etc., then you're in luck - all of the above can be controlled via HTML/JSON REST interfaces which means anything that can talk HTTP can (with the correct credentials) configure, manage, monitor your services and store/retrieve data to/from storage.
HTH.