Stranded on a tile in Forbidden Island
If a player is stranded on a lone island piece, then they can still be moved using a Helicopter Lift card. Note these can be played by anyone, on anyone's turn, to fly a stranded player to safety (e.g. to Fool's Landing).

If no one has a Helicopter Lift card by the time the island piece needs to be removed from the board, then the player drowns (and everyone loses).  On page 7 of the rules under Pawns on Flooded Tiles:


  If a pawn is on a tile that must be removed, it must immediately "swim" to an adjacent tile (up, down, left, right) that is still part of the island (even flooded). If a pawn is on a tile that is removed and cannot move to an adjacent tile, it sinks into the abyss and everyone loses the game!
  
  Exception: The Diver may swim to the nearest tile. The Explorer may swim diagonally. the Pilot may fly to any tile.


One special case to note is if the tile has to be removed and you have a helicopter card.  In this case you can save the sinking player as clarified by Matt Leacock in this Board Game Geek thread.

Note however that you may not use a Sandbags card for this purpose, as specified in the rules:


  Note: You cannot play a Sandbag card to save a flooded tile once you've revealed the matching Flood card.


Remember that all players win or lose together, so if one player drowns but the others survive then it still counts as a loss.