I found a thread on this subject that was closed last year due to age, but it didn't include an answer that works for me. It suggested restarting Firefox. Restarting Firefox, even restarting the computer, makes no difference.
I'm using Firefox 4, and just yesterday I could no longer check completed boxes. I also found when I went to the ToodleDo forum that I couldn't post a new message (I'm posting this in Chrome).