Mihai Corlan has put together a list (ever growing) of AIR applications that use a PHP backend:

I want to maintain a list with RIA applications created using Flex and PHP (could be web apps or desktop apps). While this list is unlikely to be a complete list, I hope there will be enough applications to serve as a reference point or to inspire any PHP programmer.

On the list already are apps like Shifd, Flex.org, Kizoa and a Stock Fetcher. Keep checking back on the list for more great AIR/Flex and PHP apps as they're added to the list.