Claes,
We broke it! We broke it!

I'm actually glad you mentioned "What changes are in 3.2..." initially, I would have said "Nothing of consequence".. but I'd be lying. We changed all the social modules to use classes, instead of IDs, in 3.2 for better compatibility and XHTML validation. That broke the resizing javascript we use in the Invite module.
We'll release a new version shortly, but if you can make the following change, you should be good to go. In the /modules/mod_jfbcinvite/tmpl/default.php file, on line 40, change 'class' to be 'id' and add the z-index parameter as below. Both are toward the end of the line. You can just copy this line and replace what's there if that's easier:
<div style="position: relative; top:0px; left:0px; width:<?php echo $closedWidth;?>px; overflow:hidden; z-index:100" id="jfbcinvite">
Sorry for the problems, and let us know if this fixes you up or you encounter other problems. Good luck with your launch!!