As an update to this: We're currently working on doing a code-drop for a variant of our application, a new branch of the Android app which will have some things removed from it.
It's not open source in the real definition (for now we're just doing a one time code-drop, we might do later support for it in the future) and it's not pure OSS, you won't be able to compile it from the source and get the exact same binary we have on the Play Store, but it should be an interesting experience.
I'll post more about this when more progress happens on it.
Around 90% of the mobile applications are built on top of the Stack Exchange API. The libraries we use are open source and anyone can grab them and use them.
The other 10% of the app however consists of internal things that we are not yet ready to put out to the world and say "here are all the URLs, documentation, description of what they do, go wild". This includes the server endpoints for the feed, user registration, etc. Because of these, and the fact that the rest of the apps are simple data-retrieval and demonstration, the mobile apps themselves are not going to be open sourced.
Anyone is more than welcome to use our libraries (or even make their own!) and make an application using the API, but as far as feature-requests and bugs in the official app the best thing you can do is get more votes for your requests, since questions on meta are one of our highest priority things to work on.