Origami - the Facebook prototyping tool has some amazing UI effects. Is there any way to actually use those when I'm building my app?
There's currently no way to include animations from origami into your app.
However...
Facebook announced Pop a few days ago, which is the animation engine they used in their app 'Paper'. You can hear all about it Here. And the good news is Pop integrates with Origami, making what you want possible, Pop will be open source (to be released very soon they said), so keep your eyes open for it.