Hi @David S. I´m not sure. When I was checking the code I only saw a W3 Total cache function that clears the cache. I think if it calls that function, it might call other functions from other popular caching plugins as well.
The thing is, this kind of tasks consume a lot of resources because it depends on wp cron jobs to post tweets. WordPress has to check on every page load if it´s the right time to post the next tweet and that affects the performance.
In my case, I created a custom script for posting tweets which runs on a different server. I had tweetily installed on 3 wordpress sites and since I moved to my custom script the server consumption has gone to half.