Let's say I put in a large bluff on the river in a live homegame. My opponent thinks long and hard about it and decides to make the call. I'd rather not reveal exactly what I had, knowing they are only calling if they have SOMETHING while I have literally nothing. Am I allowed to fold and forfeit my ability to win the hand?
In a tournament, no. Folding an eligible hand is not allowed. This is to prevent teams from dumping chips to each other.
In cash games, it's fine. However, a player who calls your bet is entitled to see your hand if he wishes, at his own risk. He may ask the dealer or floorman to show the hand, and if they can do so they will, but the hand will be ruled live: if you were mistaken and your "nothing" actually beats his hand, you will win. Note that this is different from the case of a third-party player asking to see a called hand because he suspects collusion: this is a breach of etiquette if abused and the floorman might choose not to do so. Also, the hand to be shown is dead. But calling a bet absolutely entitles the caller to see the bettor's hand: that's what he paid for, so there's no breach of etiquette there.
Doing this routinely opens you up to the call-bluff. If your opponent knows you're bluffing, he can call you with literally less-than-nothing knowing that you'll muck a better hand. I did this often in lowball with players who would routinely bet pairs and muck when called: I started calling with worse pairs/two pair/trips, and taking pots.
This is a fun rule that has definitely changed over time in public games, but has remained the same in home games. Amongst gentlemen, it is assumed the bettor has been caught bluffing. Demanding that a player show his cards here is tantamount to accusing him of cheating, or a blatant attempt to humiliate your opponent by showing what a dumb bluff he has made. "Gaining information" on an opponent this way has always been considered a weaselly move by experienced players.
In situations like a home game, where players provide security themselves and might very well be armed, it is a really bad idea, especially if you want to get invited back. In the past, you could be challenged to a duel and killed for less. You still deserve the same today!
Around 2003, I'm not sure exactly when [after Harrahs bought Binion's], it became a rule at the WSOP that all in players must table their cards in all-in situations. It is an incomprehensible rule, and has nothing to do with catching chip dumpers. It detracts from the strategic element of the game [by revealing bluffs], and takes away from the fraternity [by humiliating bluffers, and encouraging nit behavior]. I assure you, they do not care anything about chip dumping. If there is any sense to the rule, it may speed up the game, or it might be considered an effect of corporate politics. In casinos, management is often rewarded for "doing something". Personally, I think it is a terrible to force people to do something that was universally considered bad etiquette, just because you have a corporate monopoly and can force the issue. This is what is wrong with America. :) But I digress....
For several years, probably after 1983 and up until around 2008, and dispersed to this day around crappy poker rooms, it was a rule that players could demand to see any called live hand. Variations abounded, like only the active player, or only the calling player, or only players at the table, could demand to see the folded hand. Obviously, this caused a cascade of ruthless nit angling, and the rule was changed in Las Vegas to what it is today around 2008, I think starting with the Bellagio or the Mirage.
The modern rule in most casinos is that if you call the floor over and ask to flip up a folded betting hand, he's supposed to explain that it's considered bad etiquette unless you think that player is cheating somehow. "Do you feel the player is cheating, sir?" Whatever he says, since you don't think the player is cheating [or you would have already thrown him out], you just tell him you don't think so and walk away. Unless he's a holdout or switch artist, it's tough to see how betting a losing had and folding on the river is a cheating move, and harder see how you can "catch" somebody this way.
Ask to see a bettor's mucked hand, and reveal yourself as either a noob or a jerk.
If it's a cash game, you're under no obligation to show your hand if you don't want to win the pot. In an all-in situation in a tournament, usually the dealer would be required to turn the cards over, even if you don't want to show, but rules may vary depending on where you're playing.
Note that declining to show your hand at showdown is called "folding", just like relinquishing your hand at any other stage in the hand. "Mucking" is what the dealer does with the cards after you've folded them. If a player is able to muck their own cards, i.e., put them directly into the muck, the dealer isn't protecting the muck properly.