It is certainly possible to run CodeIgniter on Windows Azure. You will want to take care in how you manage file storage because instances do not have durable storage. Placing all non-CodeIgniter files (uploads, etc) in WAZ storage will ensure they exist
amongst all roles.
Take a look at the AzurePHP website for information on running PHP on WAZ.
The Interoperability team at Microsoft is currently looking into CodeIgniter to isolate any potential issues with running on WAZ