Imagine the following scenario: I'm playing a round in a tournament (currently Heads-Up), and my opponent folds after the turn. Can I be forced to show my cards? If so, who decides that? The dealer? Other players?
A player is never entitled to see another player's hand unless he has paid for the privilege by calling all bets. If a player folds before the betting is complete, he may certainly ask to see a hand, but that player may simply refuse.
If one or more players remain after all betting is complete, then all hands must be shown on request, though even there it is common for players to be extended the courtesy of being allowed to fold without showing.
As with all poker rules, the floorman may override if he feels it is in the interest of the game.
You can be forced to show your cards at showdown after the river, but never before that (except everyone are all-in). The moment you're the only remaining player in the hand, before showdown, you win the pot and the hand is over.
Note that you are obligated to show your cards when all players in the hand are all-in.
If you're in the hand at showdown, you can muck your hand if you don't want to show it but technically you are suppose to show it and other players at the table can ask the dealer to show your cards. It's not common but it happens.
This applies to most tournaments, cash games can have unique house rules.
if it is on the turn: let's say you bet and the opponent folds, you do not have to show your hand, ever.
players can ask each other what they had, sure. but you do not have to show as there is no official show down. especially when it's not even on the river.
it's really strange if the place you play at has a rule that says otherwise
If you win the pot due to everyone else folding, you don't have to show your cards (but may).
If you are in a showdown and your opponent presents a better hand than what you have, you only have to show your hand if you were not the last person to bet or raise (in fact, last to bet/raise has to show their cards first no matter what).
Dealer's orders are always binding though, as far as I know.