If found that checking/unchecking that "Lay out footnotes like Word 6.x/95/97/98" only temporarily restored the footnotes.
From this page, it looks like the problem may be a glich with AutoRecover (Preferences/Save). The first time after opening a document the AutoRecover feature saved the document, the footnotes disappeared (I AutoRecover set for every one minute).
As much as I feel much safer with AutoRecover on, I've since turned it off, and the footnotes are staying visible. Not sure why this bug is still present, but there you go...
I agree that this is a Word issue, but these solutions did not work for me. I have Word fof Mac 2011 V 14.3.9. What seems to have worked was inspired by DLRig's reply. I went to Preferences, Compatibility and scrolled town to find that the rule "Lay out footnotes like Word 6" was already unchecked. I checked it and unchecked it but this did not solve the problem. Scrolling down this list of rules, however, I found several that directed Word to behave like Word 2003. I unchecked all of them and so far, my notes are remaining visible and in their proper place. Turning off the AutoRecover function did not work for me.
I agree with -Sheridan- that this is an AutoRecover issue. I write a lot of Word documents with footnotes and never had the "disappearing footnote" issue until I switched to an iMac and MS Word for Mac 2011 (which I bought together in December 2013). I tried changing the footnote line spacing to "exact". I tried unchecking all of the Word compatability rules. None of that worked. And, the footnote disappearance happened, like clockwork, at the same intervals as AutoRecover was saving my document (every 10 minutes), whether I was typing anything or not. I could tell when it happened because my document text and inserted photos would suddenly shift out of place and my footnotes would disappear. (This only happens with footnotes of more than one line).
I also agree that it is safer to have AutoRecovery on, but I did not want to have to close out my document and reopen it (and realign the text) every 10 minutes so I turned it off. I hope the maintainers of MS Word for Mac will see this and fix this obvious bug.