This Twitter Timeline Feed WebPart for SharePoint 2013 is based on Twitter API version 1.1 which has many breaking changes when compared to Version 1.0 of API. You can download this free webpart here:
This webpart includes the custom Twitter feed look and the ability to customize the twitter feed to better suit your needs. To deploy the solution on your SharePoint site, download and follow these instructions:
1. Build and deploy the attached solution.
2. Activate the feature “SFS.Intranet.TwitterFeed”
3. Add the WebPart “Twitter Feed WebPart” located under “SharePoint Frontier” Category to a page
4. Edit the WebPart and go to the settings of WebPart and chose “TwitterTimeLine or Custom” as rendering mode
5. Provide values to Widget ID and Twitter Handler or all the custom fields
6. Click “Ok” to save the changes