I'm currently running the developer preview as well. And there are glitches here and there.
But this is basically something that's been mentioned allot. Twitter for some reason does not work with Android L.
Go to the Nexus 5 sub-forum then to the General section. There you will find a thread containing a list of apps that do not currently function as it should on Android L.
There is no way to get Twitter to work, you may try some of your luck and give a Twitter Client a try.
iOS7 Kik Messenger | Svelte UI CM11 Theme | Active Launcher