I've played Crysis a couple of times and encountered this bug once. I went up the waterfall and into the next level through the caves, but I knew where to go. Unfortunately I can't remember if/when the objectives display came back.
I encountered this second time through with patch 1.1 and could find no way of triggering the objective update. For me, the hostage and other nano dude wouldn't leave that room, despite all tanks being blown. Like others have said, I knew to head up the waterfall so once convinced I would not be able to trigger the objective, I finally just headed up and found my way to the cave. Surprised this still isn't fixed in patch 1.2.
Boss had the same problem, in that area and other areas, told him never to use quick save/loads and he never had it again.
yeah there's supposed to be a problem with the quicksave feature that still hasn't been fixed despite the release of two patches. You may want to replay the level on easy without using quicksave at all (though it's ok if you save manually through the menus).