Today I updated my Tweetbot desktop app on Mac to version 2.4.2. This update supports the lately introduced update to Twitter, which allows to post tweets without URLs or images counted into the 140 characters of Twitter.
After updating, the images of my accounts in the sidebar were all broken. I asked Tapbots on Twitter, but before they were able to answer, Matthias Haak replied to me with the solution, which is to remove the images in the cache via terminal with the following line:
rm -rf ~/Library/Containers/com.tapbots.TweetbotMac/Data/Library/Caches
I hope this also helps you. ;)