Here's what solved a similar issue for another client, you can check if this is relevant in your case too:
We were able to resolve the problem. It was coming from the fact that you had two Twitter Feed templates. The one had empty data in the database. Removing the template (the one that has been last modified on 17/18 September 2013) resolved the problem.
To delete the template go to Design -> Widget Templates and search for Twitter Feed. Then delete the template. Note that you should go to Modules & Services and Uninstall and Install the Feeds & Notifications module.
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