In Forbidden Island, if a player (not pilot or diver) is stranded on a tile with all surrounding tiles sunk, is the game lost or can he be rescued? Can he give away his treasure to a pilot or diver such that other players win but he (personally) dies?
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.
I don't think I agree with tttppp's answer - assuming he is saying that you can't use a Helicopter Lift in response to a tile sinking. The rules for special action cards state:
These cards are taken into your hand and can be played at any time — even on another player’s turn. Playing a Special Action card does not require an action.
Yes, there is a specific rule that says you can't use your Sandbag cards reactively to a Flood, but I don't think that implies that Helicopter cards can't be used reactively: quite the reverse if anything.
Other than that I agree with the other answers. You can't win Forbidden Island if any member of your team dies, but there are normally ways out of a tight spot... at least until the waters rise to the point of no return.