In Final Fantasy Tactics, the first plot twist, and Delita's Start of Darkness, includes a resistance knight using kidnapped "noble" (shows what he knows!) Teitra as a human shield at an abandoned fortress. Argath, under order of Ramza's Knight Templar brother Zalbag, shoots Teitra first, and then puts another arrow into the knight when he's frozen in disbelief. This gets Delita mad enough to want to kill him, and a Climax Boss battle ensues. She dies instantly, but the kidnapper actually survives, holed up in the arsenal, until after the battle where he ignites the powder stashes and blows the whole place to bits.
Goldeneye 64. A valid way of solving the hostage situation on the train level. Shoot Natalya in the leg and she lurches forward and down in pain, leaving the general open to a lead surprise.
Alternatively, you could just side-step enough so that she isn't in your way. The hostage-taker doesn't react in the slightest.
The above is because the game started out as an on-rails shooter, so the idea of moving around for a better shot wasn't part of the AI programming.
Sakura Wars 3 has a slightly odd example where the "hostage" is a painting (a very thinly-veiled Expy of the Mona Lisa). Lobelia's response? Use her pyrokinesis to destroy the painting, which turns out to be a forgery. When asked how she knew it was a fake? She didn't.
During Thane's loyalty quest in Mass Effect 2, his son, Kolyat, takes a turian politician hostage. If you don't use the Paragon Quick Time Event, you have two choices: shoot Kolyat (nonfatally) or shoot the hostage (fatally). Your possible justifications for the latter are that you kept Kolyat from doing it (the whole point of the quest) or that the hostage deserved it - we see earlier that he has a pretty drastic anti-human bias and was threatening shopkeepers.
The DLC mission Lair of the Shadow Broker has a moment when an enemy takes a hostage and makes her talk, making her confess that she has a son, she wants to live and see him again, she doesn't want to die. If you don't have a sufficient Paragon or Renegade score to talk your way out of it (in either of those paths, you can list off some of your more terrifying achievements and ask if the enemy really thinks you can't kill one innocent for the greater good), you can still choose to shoot the woman nonfatally.
Mass Effect 3 does it again when you confront Miranda's father, who's holding her sister Oriana hostage. If you can't / don't use the Charm / Intimidate options (different ways of saying "You let her go, you walk"), you get a Renegade Interrupt to shoot his hostage in the leg, giving you or Miranda the chance to finish him off.
Zoran Lazarevic, the Big Bad of Uncharted 2: Among Thieves, does this without hesitation, after giving an impassioned speech comparing himself to such "great men" as Hitler, Stalin, Genghis Khan, and Pol Pott, saying that "they had the will to do what other men would not."
Kane and Lynch. Lynch does this during one of his psychotic episodes and is genuinely distressed when Kane returns. It doesn't help that things go downhill from there as a result.
Metal Gear Solid: Psycho Mantis holds Meryl hostage by forcing her to hold a gun to her own head. One of the ways to stop her from committing suicide would be to throw a stun grenade, knocking her cold. Another way to prevent this is to punch Meryl to the point of unconsciousness. In Twin Snakes, the remake, you can also shoot her with the tranquilizer gun.
The situation repeats itself in Metal Gear Solid 4, this time with Screaming Mantis and Meryl (again).
Then there is the variation with Raiden where Vamp grabs him from behind and he uses his sword to stab himself and nail Vamp through his own gut.
The game also features an unintentional example: Hostages are immune to all of Raiden's attacks, so the easiest way to rescue them is by launching grenades directly at them to kill the hostage-takers.
The Courier is asked to do this in a side quest of Fallout: New Vegas, because the NCR troops outside Legion-captured Nelson refuse to attack as long as it would risk their comrades who are crucified in the town square. It's very possible to Take a Third Option by killing all the Legionaries and then letting down the hostages, which amazes the questgiver and gets you loyalty points with one of your possible companions.
Can be done optionally in the Modern Warfare 3 mission "Stronghold"; one of Makarov's men grabs a Czech POW and tries to take him hostage, but there's absolutely no penalty if Yuri (the player) decides to simply shoot through the prisoner to kill the thug. Captain Price even quips, "He never would have made it anyway...".
In Robopon, Mr. Wild uses a Robopon he stole from you, Dosbot, during your match with him. Before the fight, he taunts you by asking if you're going to scrap it. You have to.
In Metal Combat: Falcon's Revenge, the Final Boss shields its weak point with the ST used by your Mission Control. You can opt to destroy her mecha and take a huge chunk out of the boss's life meter, but in order to get the good ending, you have to keep her alive (a lot tougher than it looks, as the boss slowly regenerates health and hammers you with powerful attacks all the while, constantly leaving you on the defensive).
Halo 4: Spartan Ops has a variant. Covenant leader Jul 'Mdama tries to use the scientist Dr. Halsey as a shield. What he doesn't know is that Halsey has been marked for assassination, resulting his attacker shooting at them anyway. Neither of them are killed, but Halsey does lose an arm thanks to her wound.
Invoked in Star Trek Online's post-Season 8 Federation tutorial. The Player Character's commanding officer Captain Masc P. Taggart is taken captive by a Klingon battlecruiser crew. He tells the PC to lock onto his combadge signal and open fire, though the Klingon captain stabs him to death right after that.
During the ending of Batman: Arkham Knight, Batman hallucinates what would happen if the Joker infection caused him to turn as insane as the Joker was. During this hallucination, players play as Joker, carry a shotgun, and shoot several known supervillains. Riddler takes a hostage, and players can just shoot her. Riddler becomes very confused. Remember, only the players see the Joker; the villains see Batman carrying the shotgun.
In the DOS Real-Time Strategy game War Inc., kidnappers have taken a CEO of an airline hostage and made an extremely high demand. The airline considers it more profitable to disregard the safety of the CEO and simply have the kidnappers killed, while retrieving his dead body to receive insurance payments.