First and foremost, if a unit dies, except in rare circumstances in earlier games or under the Casual or Phoenix modes in more modern games, they are gone for good. Depending on the game and character, you may get to see their Famous Last Words. In games where the Player Character is an avatar you give a name to, losing someone who is a Support, or worse, your spouse, can feel as if you just lost a real best friend or significant other. More recent titles like Echoes with dubbing and even post-battle dialogue take this knife in your heart and twist it with convincing voice acting, having other characters react to the loss of their loved ones at the end of battle.
To twist the knife even further, it can happen anytime, anywhere and to anyone. It can even happen randomly in an early chapter against some random mook. Fire Emblem as a whole really shows the reality of war; that good people, people that you grow to care about, sometimes do not make it out of the conflict.
One Japanese commercial for Radiant Dawn captures the feeling of loss the game can inspire in a really understated way while a somber, bittersweet piano rendition of the Fire Emblem theme plays in the background.
99% of the death quotes are extremely sad. Intelligent Systems likes to give the goofier, funnier characters the sadder quotes.
Lex's death quote in Genealogy of the Holy War. You'd think a big tough guy like him would be going on about how a little wound's not gonna take him down, but instead?
"I'm such a weakling...Sir Sigurd, please allow me to die in peace..."
Sain from The Blazing Blade: "Lady Lyndis... I was so happy..."
And following the trend, have these from The Sacred Stones. All of these characters are among the funnier, more-stable, cheerier ones in the game, and then...
Forde: Ah, Prince Ephraim... It... It was fun while it lasted. Ewan: I'm sorry, Teacher... I never— Tana: I've long been prepared for this day... Too long... Amelia: I... I did well, didn't I? Colm: To think of it... Me... In this place... Neimi...
Probably the best example in Radiant Dawn is Kieran; you can tell by looking at him he's a very silly guy. But his death quote for Chapter 3 in Part 2?
Kieran: General... Forgive me! I should have... trained harder... Geoffrey: Kieran... No! I'm coming to help you! Hold tight! Kieran!!
Fire Emblem: Shadow Dragon and the Blade of Light: Shadow Dragon adds a special ending if Caeda dies. Marth is clearly heartbroken that Caeda died for him and realizes he loved her. and he and Nyna reflect on the curse of the Shadow Dragon. Thank Naga this ending is not canon.
Hardin. His life at Book 2 is just a massive Tear Jerker. We see his downward spiral from a just and honorable knight, who would've been Marth's greatest ally, but because of unrequited love for Nyna (who is another walking tearjerker, considering how painful her own life was as well) and Gharnef's tinkering, becomes a Fallen Hero and tyrant to his otherwise just empire. And when you kill him, he comes back to his senses, apologizes for being too weak to resist Gharnef's influence... and dies while asking Marth to take care of Nyna and asking for her forgiveness for what he did to her when Brainwashed and Crazy. Poor Hardin.
The conclusion to the above arc comes on the heels of being forced to fight and kill most of the Wolfguard, Hardin's personal platoon who were party members alongside him in the first Book. In the remake, you can save most of them, although it's implied in his ending that Wolf goes on to commit Suicide by Cop, so as the player all you can really do for him, at least, is postpone the inevitable.
New Mystery of the Emblem gives us anything and everything to do with Katarina. Though what mostly stands out would be her death quote in 16x.
The death of Joshua's mother, Queen Ismaire. Also, Lyon's conversations with Ephraim's are even more heartwretching than the ones he has with Eirika. The trick is how different the twins' reactions to Lyon are; Eirika has always been emotional and that's okay, but seeing the calm and collected Ephraim lose it when he faces Lyon...
"C'mon, Ephraim. Smile for me, like you always do..."
Knoll and Duessel's A support conversation: "We are dead men who strayed too far from our graves." Knoll himself is a magnet for tragedy, especially his ending. He doesn't even stay to enjoy Grado being back to normal, and if you go by the aforementioned support, it's possible that he's dead.
Actually his ending and supports with Natasha implies that he stayed and helped restore Grado, and eventually left his homeland, likely out of guilt for being one of the assistants of Lyon who unintentionally assisted his prince in causing a war and resurrecting the Demon King, which is also quite depressing.
The very last piece of dialogue in the game, a post-credits picture of Lyon meeting Eirika and Ephraim for the first time. The text box only gives Lyon's speech, and it rends you to the core after the ending you've just been through:
Lyon: Oh, are you... I'm... My name is Lyon. Uh-huh. That's me...The son of the emperor. You're Prince Ephraim and Princess Eirika from Renais, aren't you? I heard you were coming, and I've been waiting here all day. I've always wanted... I don't have any friends my own age... So, Prince Ephraim... Princess Eirika... What do you say? From now on, let's be friends...
A rather subtle one: if you watch the portrait sprite of the character during their death quote, the character actually closes their eyes after the quote finishes (except for people who just retreat).