I can confirm the behavior, if you click Add Reply without checking off the reCaptcha box. When the postback returns the page, the reCaptcha box is missing.
However, to get it back, just click in the address bar and press enter. It does a new request to the page instead of a refresh, which also posts data. When you reload it, you have to avoid the post. When you do this, copy your reply out so you don't lose the text.
This isn't a solution, just a workaround, but at least it should prevent you from having to close and re-open the browser.
Also, I use PaleMoon, a Mozilla branch, so I am not using a straight Moz. FireFox branch either.