Dub Text in Digimon Tamers final episode. When the program that was used to defeat the D-Reaper forces the digimon to return to the digital world, Henry's dad, who knew it would happen, asks him if he can forgive him, and Henry sadly shakes his head. (In the original, the head shake meant "don't worry about it").
Perhaps one of the most heartbreaking example was in Kung Fu Panda, where during the climatic battle between Shifu and Tai Lung, Shifu apologizes to Tai Lung for allowing him to go down his path of darkness, only for Tai Lung to reject it and demand the dragon scroll.
In The Incredibles, Mr. Incredible apologizes to Syndrome for the slight that put him on the path to supervillainy. Syndrome answers "See? Now you respect me. Because I'm a threat," and continues monologuing. Mr. Incredible realizes that Syndrome can't be reasoned with, and starts throwing stuff at him.
In Pitch Perfect, Beca, the protagonist goes all the way up to her love interest to apologize for being a Jerk Ass earlier. The love interest curiously, doesn't accept or even appears to want one, instead pointing out her flaws in driving people away.
In the 2007 film Bridge to Terabithia, Jess Aarons tried to apologize with his younger sister, Mary Belle for pushing her because he was angry by the death of Leslie Burke and told her he misses her talking to him, only for Mary Belle to ignore his apology by turning away from him.
In one episode of The Big Bang Theory, Sheldon accidentally ruins Howard's chances to work on a space laser. Feeling guilty, Sheldon apologizes, only to find out that Howard isn't so willing to forgive.
After Frasier convinces Roz to ask someone out which ends in disaster, he asks before an ambitious broadcast for her forgiveness. Roz sarcastically complies, especially, as she adds, under the thought he will out there on air humiliating himself in front of thousands.
Frasier: So when are you really going to forgive me?
Roz: When I get married!
Also happens when Frasier is caught looking in Daphne's room, realizing she is strongly protective of her privacy, which ends with him having to bribe her with a car so she doesn't quit. After he inadvertently meddles in her belongings in a later episode, it seems the process will repeat, until a furious Daphne accidentally reveals she was snooping through Frasier's own belongings in a rant. After being called out on this, a flustered Daphne decides she'll let it go this time.
In the Babylon 5 episode "Comes the Inquisitor", Vir ends up on an elevator with G'Kar, and tries to apologize for the Orbital Bombardment on the Narn homeworld by the Centauri, which occured the previous episode.
Vir: "I'm sorry. I wish... there was something that I could do. I tried telling them, but they wouldn't listen. They never listen... I'm sorry."
G'Kar(pulls out a knife and cuts his own hand, and as drops of blood drip to the floor, says): "Dead... dead, dead, dead, dead, dead, dead, dead, dead, dead... How do you apologize to them?"
Vir: "I can't."
G'Kar: "Then I cannot forgive."
In the episode "Ship of Tears", when Delenn finally brings G'Kar into the Conspiracy of Light and discloses that she knew all along about the Shadows and their aid to races like the Centauri--but explained to him that consequences and casualties would have been much worse than what befell his homeworld had they took action and shown their hand against them too early--G'Kar did understand but was still angry that millions on his homeworld were sacrificed. Delenn asks if he'll ever find it in his heart to forgive her, and tearfully he replies:
"Perhaps, but not today."
In Fringe, Walter's halting attempt at apologizing to Astrid for injecting her and knocking her out is met by her completely ignoring him.
In the Friends episode "The One With The Baby Shower", Monica forgets to invite Rachel's mother to Rachel's baby shower, and spends the whole episode desperately trying to have her apology accepted.
In Star Trek: Deep Space Nine, Odo tries making a verbal apology to Kira after he breaks with the Female Changeling, under whose influence he abandoned their resistance, leading to Rom's arrest, impending execution, and looming disaster for the Alpha Quadrant. Not surprisingly, she tells him that are "way, way past sorry." (He does earn her forgiveness for helping them fight off the Dominion and a long conversation some episodes later.)
One Republic's song "Apologize" is based ENTIRELY on this trope.
The Bible: Samuel rebukes King Saul for disobeying God's commands in destroying every single Amalekite and only spared the king and their sheep and cattle. Saul sincerely repents to the LORD, but Samuel tells him that God won't accept it and He has rejected him as king of Israel.
In an episode of South Park Cartman infects Kyle with AIDS for laughing at him, leaving the latter obviously enraged. The principal finds out about this and demands Cartman give an apology (which he does half heartedly), while Kyle should also apologize for tattling on Cartman. Cartman seems sincerely appalled that Kyle is still livid.
One episode of American Dad! has Stan taking his family to his half brother Rusty's place, only to discover he's fabulously wealthy. Becoming jealous, Stan and his family attempt to take Rusty's place, only for them to be forcibly sent off with a threat that he'll kill Stan if he ever sets foot on his land again. The helicopter they're in then crashes and they're stranded in the desert. After going through hell and realizing that family is more important than material posessions, Stan attempts to make amends with Rusty who found them in desert, only for his half brother to attempt to carry out his threat of killing Stan for still being on his land (he owned alot apparently).
In a rare hero-to-villain case, Danny tried to apologize to Vlad for pranks he pulled on him after Vlad began making life miserable for him and his friends. Unfortunately, Vlad wouldn't have it, made a big speech that Danny deserves what he's getting, and sics the Guys In White on him.
In the Family Guy episode "Movin' Out (Brian's Song)", Brian wants to apologize to Jillian when Stewie revealed that he was paying half of the rent of the apartment, which prompts Brian to admit that he never wanted to move in with Jillian at all. She still rejects his apology and remains broken-hearted.
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