For some reason I'm not able to share any posts from Facebook onto my own Facebook wall using my iPhone 5s, but I am able to do it with my PC and laptop. If there some special setting I need to know for my iPhone?
I agree with @youngjm. The Facebook app is so buggy that it's hardly worth using at all most of the time. Here's a tip: try accessing your Facebook account from the mobile site using Safari on your iPhone. If you are able to successfully post via the mobile site, then it is definitely an issue with the Facebook application and not your phone.
I was having the same problem with my iPhone 5s and Facebook. I unistalled the app and reinstalled it and it has worked (so far). I agree with the other posters that it is a buggy app. The other thing you can try is going to settings on your iPhone and scroll down to the Facebook setting and make sure everything is ok there.