As pointed out by iMore, the Facebook iOS app added functionality allowing you to save posts to a new “Saved” folder. Others are not notified when you save posts, and posts must be unsaved to be removed from the folder. The new feature does not require an update from the App Store (it is also rolling out on Facebook.com), but Facebook will notify you in the timeline.
To access your Saved folder, you must first scroll to the bottom of the sidebar menu and tap on Edit Favorites. Next, tap the + sign and select Saved from your list of Apps. Then tap Done. Now, you should see your Saved folder in the Favorites section of your sidebar menu!