Have you tried clearing your cookies and history?
That's got me thinking.
The forum software (somewhere) remembers what you've typed in the reply box until you hit the post button. As I mentioned, and to demonstrate, start typing your reply in one browser then switch to another browser (same thread obviously) and the new browser will pick up what you've already written, allowing you to edit and post.
So the forum software is somewhere storing an amount of 'in process' history, albeit for a brief period of time, until the 'post' button is hit. I noted that the font of the reply sometimes changes, and it seemed to me that that was when the 'post reply' button disappeared too.
If I were the programmer, I'd have a look at what happens around that 'refresh' of the reply box. It could be triggering something (in some browsers) that's causing the post button to disappear. Dunno. Sorry, I was a coder so I tend to think in those terms.