NPCs are demanding people. They will ask you to donate money, give you Pop Quizzes about what you've done on your quest, send you off on annoying sidequests, make you run all the way around the world for that useless puzzle item or trading game item... and the hero will, of course, be obliged to do as the NPC says, because, well, good guys like you are just into philanthropy like that (not to mention the fact that it's necessary for your quest half of the time.)
Sometimes, though, an NPC will go so far as to demand that you hand over one or more of a useful and valuable item from your inventory. These aren't puzzle items whose only existence in life is to appease them. Rather, they're items that you bought with your own hard-earned gold and actually intended to use on yourself. Surprise, now you're going to have to buy another one.
When you hear the NPC say, "I need your life potion to heal my daughter" or "I need some bombs to blow up this wall," you know that the item is going to vanish from your inventory and you're never going to see it again. At least giving the item to the guy will make him happy enough to let you pass, or maybe give you that useless puzzle item you need to give to the king of some other country...
This often results in an the player getting something much better in return, sometimes later in the game, basically making it an unspoken Aesop about giving stuff to people in need.
The most famous example of this comes from The Legend of Zelda. In Dungeon 7, there is a monster that will stand in your path and say, "Grumble, Grumble..." until you hand it your bait, a normally useful item for luring enemies away from you.
And finally, there are a few dungeons in the second quest with some very evil old men hiding in them. When you enter, the doors will lock and they will demand you either give up some money or a Heart Container. Yes, that's right: if you don't pony up the cash, you lose one heart off your maximum life. Forever.
Also from Zelda, the well trading sequence in Majora's Mask is based on useful inventory items instead of the usual event items. Tip: Don't go in there without bringing a blue potion and many seeds.
Majora's Mask has two occasions wherein an NPC will ask for a health potion. The first, required to advance the plot, is for the injured witch Kotake, though you'll get the potion for free if you go tell her sister; the other is for an invisible soldier who will thank you by giving his stone mask.
The Wind Waker, meanwhile, has the Elixir Soup item, which requires you to use a fairy on your grandmother instead of yourself in order to get it. Given that the soup does most of what a fairy does, only better, this isn't a big problem.
In Link's Awakening, at one point late in the game a monster can be found in a cave on the beach who claims to have found a special item, but will only give it to you if you give up the item in your B slot. Give up an item you don't need anymore (like the Shovel) and you get the Boomerang, which is DEFINITELY a special item.
In Oracle Of Ages, the magical sickness Veran inflicted on King Zora in the distant past can only be healed by a red potion. Yes, that red potion.
The "wimpy" knight (who cannot lift and throw a barrel) in Skyward Sword makes you give him a stamina potion so he can do push-ups (he can only manage a few at a time). A while later...he makes you give him another stamina potion. But a while after that, he's able to lift a barrel with one hand, does thousands of pushups in a rowas a warmup, and lets you play a minigame where he hurls large pumpkins into the air so you can shoot arrows at them.
In the original Animal Crossing, the residents of your town would occasionally force you to hand over one of the items in your inventory in exchange for something random, but generally of low value if you talked to them. The later games removed this, presumably due to how annoying it was to have to store all of your valuables before interacting with any townspeople.
In Baldur's Gate, Branwen is a priestess turned to stone by an evil mage. You can buy a stone to flesh scroll from the douchebag who's exploiting her situation for 500 gold. Or, you can go to a temple, get a scroll directly from a priest, and use that instead (and unless you've been butchering innocents along the way, you'll probably be able to get it for less). Or or, you can knock said douchebag over the head, and take his scroll for nothing.
The monkey cave in EarthBound has a similar structure as Majora's Mask's well.
In the game Faxanadu, you must revive three fountains to obtain passage to Mascon, the misty area. Reviving one of them requires you to give up an Elixir, the game's Auto Revive item.
During the Colosseum event in Breath of Fire II, Ryu has to give Rand all of his money. Thankfully, you can bank your money beforehand, and dutifully give Rand your nothing with no ill effect.
In the SNES quasi-RTS OgreBattle: The March of the Black Queen, episode IV, March of the Black Queen, you will have NPC characters that require fetch quests, lengthy Chain of Deals for more items, and at least one deal that involves your hero's Sword of Virtue. (And if your hero is less than pious, the demon will join your side.)
The Magic Pots of the Final Fantasy series give good experience at the expense of one or more elixirs, the most expensive and rare health item in the series. When you fight them, they exclaim, "Gimme an elixir", and when you do, they run away, leaving you with a generous helping of experience points.
Near the beginning of Final Fantasy III (NES/DS), you find the first of the game's rare and highly useful "Elixir" items. About one town over from the guy who demands one for his sick wife. Then you have to do this two or three times more later on.
In Final Fantasy IV (The DS version at least), Namingway (or whatever he was called at that point) asks you for a tent. This doesn't seem like too great a request, until he thanks you for giving him ALL of your tents.
In Final Fantasy IV: The After Years, Porom's challenge dungeon is full of this. The problem comes when you realize you're timed, and you've only got one good fighter on your team, so a smart player will be running away from all battles... which means that said player won't have the money necessary to buy much of the expensive armor that the NPCs tend to request, especially since running away from battles always has a decent chance of you dropping 25% as much Gil as you would've gotten from winning that battle. Just one of the many reasons Porom's challenge dungeon is a Scrappy Level.
The Magic Pots in Final Fantasy V give out one hundred ability points, very useful at that stage of the game, however due to the amount of required elixirs being randomized you may end up losing them all and gaining nothing.
The GBA remake of Final Fantasy VI has a malicious variant called a Glutturn that demands an item, and if not promptly given said item, will proceed to assault the party with some downright deadly attacks, and will unleash an even nastier one upon its demise. Killing it or giving it an item results in zero rewards, but additional copies of some of the game's best accessories can be stolen from them.
Reversed in Crisis Core which, after feeding the Magic Pot exactly what it wants (in this case, specific attacks/magic rather than Elixirs), allows you to summon it in battle as one of the six Chocobo Mode DMWs. Doing so allows you to help yourself to its inventory for a change. Encountering it again and using the attacks it wants also gives you 1 out of 2 rare items depending on what mission you find it in, the last one being required for one of the pieces of Armor of Invincibility.
One of the more important sidequests in Final Fantasy IX was giving certain "friendly" creatures the specific gems they ask for. You're awarded a considerable amount of AP (requiring for mastering abilities), a clue to the next friendly in the chain, and in the later part of the chain, the gem that the next creature will want. Meeting and giving all of them what they want makes the Bonus Boss easier by making it targetable by normal attacks and making it take damage from one of its own attacks instead of getting healed by it.
Also of note is "Gimme Cat", a normal monster that exploits this trope by trying to fool you into giving it a diamond (note: this is a game where other gems are thrown at you by the bucket load, while you'll be happy to ever actually find one single diamond). Give it the item and it runs away. However, the fact it uses the normal battle music should clue you in about the deception attempt.
In Final Fantasy XII where it was possible to actually steal your Elixir back after giving it to the monster or even kill it instead of handing it over by using defense-ignoring weapons and technicks.
Final Fantasy Tactics A2 quests usually ask for loot, which is otherwise only good for Item Crafting anyway, but there is one repeatable quest that asks for an ether, easily one of the most valuable items in the game. And for extra fun, the NPC is never found at the appointed place so you need to run after him.
It also has Magic Pots as monsters, who'll chase your units around and keep yelling "Gimme Elixir!" at them until you listen, attack them, or complete the mission. Giving them an Elixir only gets you a low value item, and attacking them makes them start OHKOing your units, so simply ignoring them is the best course of action.
Freeware game God of Thunder had a few "give me something" characters as well. Sometimes you get a hint to continue the game, sometimes an item, but one simply says "Thanks" when you give 100 jewels.
Subverted in The 7th Saga: a girl asks for a topaz (you sell for money), in exchange for a secret. When you give her... she tells you she'll get married. And laughs at your face.
Nearly everything in Arcanum is absolutely persistent, though; you just have to break out of the traditional RPG mindset to take advantage of this fact. The guy who wants your camera? You can give it to him and lose an opportunity later, or do the standard RPG quest to find the information you need to continue. Or you can give him the camera then pickpocket it back from him later. You can also let yourself into the back room of his store by either picking the lock or pickpocketing the key off him and finding the information you need in his records. (You can also help yourself to his store inventory in this room, which means you can steal all of his gold by stealing everything he has for sale, selling it to him, then walking into his back room to steal it back and sell it back to him again.) These alternate solutions are facilitated by the fact that this particular NPC has really crap perception skills compared to most, especially other shopkeepers.
In Sword Of Vermilion, at one point halfway through the game you run into a certain seemingly typical shopkeeper who sells three joke items, and upon attempting to purchase any of these, he claims you don't have enough money, takes all of it, and compensates himself for the rest by taking all of your swords, too; in order to not leave you entirely unarmed, he'll hand over a crappy early game sword, leaving you without money to easily replace it with. Close to endgame, your eventual reward for this stomach punch is the titular Sword of Vermillion, the most powerful sword in the game. It's also possible to avoid the shopkeeper entirely and prevent the loss of all your hard earned stuff, though you do miss out on the sword reward as a result. Besides, a decent amount of money can be spared by buying other expensive inventory items like Shields and Spellbooks, then reselling them afterward.
Averted in The Elder Scrolls IV: Oblivion, wherein various highwaymen will demand that you give them some gold as a threat. Refuse, and they try to kill you. Hm... You do get some nifty gear from their corpses, though.
The first expansion of The Elder Scrolls III: Morrowind has Gaenor, who asks you for increasingly large sums of money (and/or rare items) until he refuses to believe that you'd actually have the cash/item and storms off even if you have the cash.
He then shows up a few days later wearing a full set of one of the most powerful and rare armor sets in the game and tries to kill you - and is nearly invincible due to said armor and his cosmically high Luck score.
A tricky example in The Witcher. The player comes across a lady who needs a potion to ease her son's slow, painful death. However you cannot commonly buy such potions in this game, but you can make them. This requires meditating for a certain amount of time, but once you've done this there's a good chance that the woman will have disappeared, never to be seen again.
Somewhat less troublesome in that it's an easy to make potion base that you use frequently in the game and more often than not costs you nothing to brew (alcohol is easy to find), so not having several in your invetory already is at least silly.
Ōkami does this once or twice (most of the time it's puzzle items) but as Ammy gets stronger with praise, it makes sense for her to give in to the demands of the little people.
Mega Man Battle Network is horrid with this. There's at least one quest in every game that makes you go haring off to the dark corners of the net and back again (at least for that point in the game) just for the bleeding chip an NPC wants. Sometimes you need to do this to advance the plot, mostly blatantly in 3, other times it's just on optional fetch quests, but it's almost always something you'd want to use yourself. At least they don't ask for any of your Power Ups...
Spoofed in Battle Network 2. An NPC, in exchange for an important tool, demands all of your HP and powerups. After the exchange, he reveals that it was a Secret Test of Character to make sure you were sincere about saving the internet, and you leave none the wiser (or weaker).
Kingdom Hearts I has this in Wonderland, where you have to give the flowers an Elixir. In return, they'll either give you items or make you return to your normal size. The one which changes your size is necessary to finish the plot, but the rest are usually worth it, since Elixirs can be picked up by destroying enemies fairly easily. It only works once (except for the size change), but the flowers will accept Elixirs and return nothing, so make sure you remember which flowers you've hit up!
Present in Kingdom Hearts 3D: Dream Drop Distance, where the abilities on the Ducky Goose and Drak Quack's link boards can only be accessed by sacrificing items, ranging from common to very valuable. To compensate, the amount of link points needed to actually activate their abilities is much lower then that of other Dream Eaters.
Kingdom of Loathing has an optional quest where you bring a guard a healing potion. Said healing potion is cheap, and the guard gives you the money for it. Indecently, he gives you the money first, bringing back the potion is optional.
This also introduces the resident healer (Doc Galaktik) to new players who might not have thoroughly explored the town.
Atlantica Online does this quite often, but in many (and sometimes unexpected) variations. In at least one case, you bring the item only to have the NPC respond he actually doesn't need it anymore, and gives it back to you and suggests using it for some Item Crafting. In other cases, a quest may ask you to craft something, but you don't need to give it to the NPC as proof.
Deus Ex has a few examples of this. If you achieve the side-objective of rescuing fellow agent Gunther in the first mission, he will ask you to lend him a pistol before makes his escape. If you don't give it to him, he'll be much less friendly the next time you see him as he thinks you (the newer model of cyborg) are looking down on him.
You can also give food and money to certain beggars on the street, which will result in them telling you about secret entrances into the place you are trying to infiltrate.
In an unexpected turn, there's a part where you meet a pilot in a bar who asks for a beer before he'll talk to you. If you hand over a beer you've been carrying round since before this mission (or pick one up from a random location) started he'll complain that it's warm, meaning you have to go over to the bartender and buy a new one.
Late in Deus Ex: Human Revolution, a friendly hacker will ask for a gun to defend himself against the attacking mercenaries. If you comply, he will find you later to reward you with a nifty amount of cash; if you don't, he dies soon thereafter. The problem is, your inventory is very limited, and you are not likely to carry around a spare firearm to give away. Thankfully, there is weapons vendor just downstairs from there, so reloading a saved game and a short detour fix that.
Phantasy Star has a number of beggars who'll tell you something about the game world in exchange for a cola healing item. Be warned that there's one who will take your cola without telling you anything at all!
Pokémon Red and Blue/Yellow required you at one point to bribe a guard by offering him one of the three "beverage" healing items (Fresh Water, Soda Pop, or Lemonade) purchasable in Celadon City. Remakes FireRed/LeafGreen changed this to a standard Fetch Quest instead, because it used modern mechanics which allowed for non-Key Items to be traded, which would allow access way earlier than intended.
Pokémon Mystery Dungeon has this for nearly half the rescue jobs you take, hunting down a character in a dungeon just so you can give them something.
Averted in The World Ends with You, where you need only show them whatever item they request, and get to keep it afterwards. In cases where it's directly implied they want a food item so they can eat it, they instantly decide they look/taste disgusting and give it back.
The Disgaea series has the Dark Assembly, where you have NPC senators vote on various proposals you put forth (for effects like unlocking new character classes, making the enemies stronger or getting more EXP from the next thing you kill.) The standard way to make the senators more likely to vote in favor of your proposal is to bribe them by giving them items from your inventory. Inverted slightly in Disgaea 2 and 3, where under certain conditions you can make the senators give you things from their inventory - though usually by the time you can do this, their stuff is pretty worthless.
The main challenge in the bribing is not getting the items (they all are sold in a shop just outside and you can use the cheapest varieties) - it's preparing all kinds of items beforehand and then bribing the right senators.
Occasionally in Fallout 3, you'll find a guy just setting around in the Wasteland, begging for Purified Water, a rare commodity in Post-Apocalyptic America, and you can give him yours. The only reward for this is a few points of Karma, and indulging in Video Game Caring Potential
Purified water is available for free from your robotic butler.
Oh, also, if you agree to help the Outcasts by Fort Independence find new technology and have Outcast Power armor in your inventory, the jackass leader will notice it and immediately take it from you. Since he assumesyou looted itoff his men's corpse before they could recover it, and doesn't do the same with the one unique set that you legitimately earn. He also doesn't press the issue if you've helped them enough to be considered friendly (by donating enough pre-War goodies in exchange for a choice of a couple of useful commodities).
All Fallout games have a few sidequests that require you to give an NPC a tool or somesuch item. They're divided into essential parts that you might have picked up (or else return once you find one), and those that let you complete a step of the quest quicker.
In chapter 4 of Super Paper Mario, Squirps, the door-opening, assistant character of that chapter, will at one point go into a door with two 'Squirps-holes' (for lack of a better explanation), and then stay there, refusing to move unless you give him something to eat. Unless you know what to do, most players will often give him something out of their inventory, like a Mushroom. Bad idea. Squirps will kick away any item which isn't a chocolate bar (which leads to you going on a 'small journey' to find one), and will stay there, refusing to move on. What really is the icing on the cake though, is that Squirps does a So Long, and Thanks for All the Gear on you, by taking away your main form of attack (him).
In Paper Mario and its sequel, Koopa Koot and Puni the Elder (respectively) ask for a fairly elusive and costly Life Shroom. At least in the latter case, you are rewarded with a fairly large coin bonus.
Avalon Code has a variant of this as well, in that you can offer "presents" from your Book to complete some sidequests and help with character relationships. Unlike most examples, you get to keep whatever it was you just gave out, even if it's your Infinity+1 Sword, since you can produce as many copies of them as your MP allows..
In Tales of Symphonia, the party encounters a group of people and one of them walks into Colette, dropping a bottle of potion. They get all pissy and demand you go buy them a new one and won't let you leave. The party will eventually find out the other group is in fact impersonating the protagonists in order to cheat money and items from people, including a unique book they needed from the local governor. It is possible to learn all this before giving them the potion, but you are still not allowed to refuse.
There's another point in the game where you encounter a random little gnome in a dungeon. He wants a potion and unless you give him one he won't return to the Earth Temple. While you can easily proceed through this level without giving it to him then, until you send him back you won't be able to enter the Earth Temple, which you will eventually have to do. Also, in the Earth Temple, you will have cook Curry for the gnomes, and you do need to have the ingredients to cook it.
In Romancing Sa Ga: Minstrel Song, beggars and thugs dot the streets of South Estamir, and will chase you down if they spot you. Beggars plead for a handout, while thugs just demand all your money. If you agree, the kids give you information in return for a little cash, while the thugs... just take all your money. Refusing, on the other hand, makes the beggars insult you and the thugs attack. Pretty annoying.
In The Legend of Kyrandia III: Malcolm's Revenge, you encounter a pirate who takes you prisoner and demands that you give him valuable items from your inventory. By this point in the game you have likely picked up a fair quantity of gemstones, and if you don't know the secret to getting past this part, you can end up giving them all to him without satisfying his demand. Instead, you have to give him the gold collar you acquired in the previous section, which he will promptly put on — only to be yanked away by another antagonist, leaving you free to go.
In Dubloon, one NPC on Stern Island demands you to give him all your rum; a revive item, in an exchange for important item.
Resident Evil 5 allows the players to do this to each other in co-op mode! Opening up your inventory has a button to view your partner's inventory, and selecting an item there will cause your character to call out, "Gimme an egg!" or "Gimme a grenade!"
In Magician's Quest: Mysterious Times, various special quest characters will frequently ask for items. For example, a yeti who wants to have a feast might ask for a specific fish, or a dragon who's sick will ask for Dragon Scales (a magic ingredient) to heal himself.
In Monster Rancher DS, the first time you go to the Kawrea Volcano errantry, you will find a burnt-up tree with a single Golden Peach on it. The Golden Peach is one of the most useful items—it allows you to add an entire year to the lifespan of any monster—but the first time, you're supposed to use the Peach to heal an injured Phoenix. You could abscond with it and use it on your monster, but... are you really so heartless?
Appears throughout the Etrian Odyssey series, both in and out of the dungeon. For instance, in Heroes of Lagaard you can find an exhausted, stranded guard who begs you for a Warp Wire.
There are a couple of these in Dragon Age: Origins. In Lothering, three separate side quests ask you to hand over three simple health potions, poisons, and traps—all products of the game's three crafting skills that you may or may not have learned by that point. The Blackstone Irregulars want health poultices, the Mages' Collective wants you to bribe the Templars with some lyrium potions, and a shady character wants garnets. Also, the armies you gather to help you end the blight each ask for some variety of supplies; dwarves ask for gems, mages ask for runes, etc.
Divinity 2 has a quest that does this with stat points as the given item. The sacrifice is hardly even noticeable, but they are stat points. That's like the RPG version of organ donation.
In order to get your Optional Party Member in Odium, you have to use a stun grenade on the door behind which he is hiding to force him out. Not exactly "giving" per se, but works all the same.
This comes up occasionally in the Mass Effect games. From time to time, you'll encounter injured NPCs. In exchange for a few Paragon points and information of questionable usefulness, you can give them medigels. This will sometimes be lampshaded by your more Renegade aligned squadmates to the effect of "Why are we bothering?".
In Yakuza 4 you sometimes run into people who ask for a specific type of item. Helping out the one homeless guy asking for booze or the other homeless guy asking for any drink at all results in a new friend who may randomly come help you out in a fight.
While most World of Warcraft quests require items that can only be obtained while the player has the quests, some require the players to provide actual inventory items they could otherwise use. Before each capital city had tabards, the main way to get reputation was to turn in large amounts of cloth. Learning how to smelt dark iron or become a Goblin Engineer will cost you regular goods. Some dailies require players to craft items (food or gems/jewelry) and turn them over, but generally compensate well enough even to justify buying ingredients if necessary.