I have the same problem. I'm using the latest version of add this for iOS6 (addthisios-0.2.0c). It also happened with the previous version of AddThis I was using on iOS5.
My app uses twitter, facebook and linkedIn but it also happens when I try with tumblr or delicious, after being in the browser view and either posting the shared item or cancelling the action. When I return to my app from the browser view nothing is interactive. Like the original poster says, there is no crash. It's like there's a transparent view covering the entire screen that blocks user interactivity.
I have tried both the showAddThisButtonInView: method and the single service shareURL: method with the same results.