@senols: Tweet Blender caches tweets by using a centralized cache system that is maintained by ALL visitors to your site. This allows visitors that reached their Twitter API limit to still get fresh tweets because other visitors update the cache for them.
Each widget sends a cache update request from the browser. If user opened your site with the widget on it and never closed that browser window the widget will keep refreshing and trying to update cache. If you delete the plugin the browser would still keep trying and you'll see the 404s you are seeing. Until that browser window is closed or page is refreshed without the widget in it you will still see cache update requests.
If you have a high traffic site, use many different widgets with different settings, or put in many sources in tweet blender then the cache update requests will be more frequent and can overload your server. You can:
1) turn off caching completely by going to WP Admin > Settings > Tweet Blender > Archive tab
2) Increase your widget refresh rates so they refresh less frequently
3) Use less individual sources e.g. create a twitter list, add all users you'd like to include in your stream to that list, and then use that list as the only source for the widget.
P.S. If your website's uptime is important I recommend you test the plugins in a test environment before putting them on the live site.