First make sure you run AJAX Scroller 2.5.1 or later. Older ones use old version of API that Twitter retired.
Twitter API allows clients to make limited number of requests per hour. In order to keep AJAX Scroller below this limit activate caching in parameters (AJAX Scroller module > General Settings > Cache), set cache time to 1 minute.
Also make sure your Joomla caching works correctly and you don't have any permission problems. You can do so refreshing your frontend and going to Site > Maintenance > Clear Cache. You should see mod_ajaxscroller Cache Group and number of files created should be greater than 0.