Is it possible to lose Forbidden Island on the first turn?
Starting off with Fools' Landing being flooded is certainly a set-back, but all is not lost! 

Naturally, due to its crucial nature, if Fools' Landing does become flooded, shoring it up becomes a top priority. Your first option is to immediately move adjacent to Fools' Landing, if you can, and use an action to shore up. Failing that, any player can play a 'Sandbags' card to shore it up as a free action.

If, sadly, you can't move that far and no-one has any sandbags in their starting hand (as can happen easily in a two player game) - Fools' Landing is left somewhat vulnerable to a Waters Rise card, but even then it's not guaranteed that Fools' Landing is re-drawn and sunk.

Fortunately, it's pretty rare that you start too far away and have no sandbags and Waters Rise and Fools Landing is drawn - but if it does happen, take solace that your end was a quick one, and hope for better fortune next time.