There are a lot of posts in forums and blogs about caret going missing (becoming invisible) in Firefox (<input> text and password, textarea). It is supposedly fixed in Firefox 3, but still happened to me with FF3 beta 5 and my Estonian-English-Estonian dictionary.
None of the fixes worked for me, cursor was still missing. However, I tracked the bug to disabling of the editbox – if it wasn’t disabled during AJAX query, then the cursor did not disappear. Long story short – it seems that the issue only occurred when editbox was the active control (i.e., caret was in the editbox). So, the solution that worked for me was simply to move caret away before disabling the form elements (element.Blur;):
This may or may not work for you – but it worked for me.