Bernado Nguyen-Hoan has found an easy to follow method for using the Trending Tags Web Part outside of My Sites. You can view his post here:
The one challenge I faced was that since I needed to work in SharePoint 2013 Online, and am hesitant about activating the My Site Host feature, I would get errors when someone clicked on a hashtag related to a missing display template.
However, Bernado had an excellent suggestion of copying the Control_Tagfeed.html display template from a My Site and uploading this to the non-My Site destination I needed to use it in. The display template is found in the Master Page Gallery via the following path:
Site Settings > Web Designer Galleries > Master Pages > Display Templates > System