I just registered on this forum for this thread only, because I can't find any other info on it on the web.
I seem to have exactly the same problem with Touch ID after I have installed Twitter, it seems. I have not used twitter since I got my new iPhone 5S, which I got at the start of November. No problems at all. Then slowly sometime in December the problems started to occur, where Touch ID wouldn't work in a way, that when I used an unregistered finger, it won't let me in by showing the image of 4 vibrating dots, but when I used a registered finger, it did nothing. So this means, that since it knows - in that moment - what finger not to get access, it most surely also must know the one(s) it has to give access - I presume - but does nothing.. As I haven't used the right fingers on the sensor at all, only the wrong/unregistered ones.
Also, sometime in December I installed and started using the official Twitter client, so I'm highly inclined to believe that this is (also for me) the one causing the issues. Also, there have to pass some time after someone locks the phone in the Twitter app (or maybe also Twitter running in the background), before it can fail in this way, otherwise, it works fine, if not enough time has passed.
Running iOS 7.0.2 (of fear upgrading and getting even more bugs

). This bug is also the only reason I haven't jailbreaked yet, so now - thanks to this thread - I will now try to install TweetBot, uninstall the official Twitter client, and see how it will go.
Also, I think that one of the reasons that makes this bug possible to happen with the official Twitter client is that tweets are an integrated part of iOS, and that, on a deep level and when the official Twitter client is installed, iOS registers its presence. If the bug goes away with the use of unofficial Twitter clients could be, that there is no longer an iOS connection to the twitter client in the same way. And when there is, like tweeting from Siri or Safari, it closes down after a tweet is sent. But I don't tweet, only read on it, so I have no idea how like Siri tweets or reads tweets, with an integrated client in iOS itself, even without any client being present, or a client have to be present, but only for cosmetics sake, and iOS still uses its own built in client, or if it actually uses some functionality due to the user installing a client. If I remember correctly, you have to have the official client installed for like tweet from Safari or Siri, but - of course - doesn't have to use it for client, and can just use a 3rd party one.