As Ryuko is falling back to Earth in the final episode of Kill la Kill, Satsuki runs towards her. There was originally no dialogue, but Satsuki's voice actor felt this was out of place, so she screamed "RYUKO!" as Satsuki ran. The directors felt it added something to the scene, and left it in.
It's not dialogue, but when Digimon Tamers head writer Chiaki Konaka saw that Jianliang's little sister, Xiaochun, was included in the opening song's shot of characters holding up their Digivices, he decided to make her a Tamer. Similarly, when he saw that the character designer had drawn so many pictures of her infamous "Terriermon torture", he decided "it would be criminal" not to include scenes of it. Xiaochun and Terriermon even have an Image Song together.
Konaka also states that Kiyoyuki Yanada (Guardromon) frequently adlibbed lines during recording, for which he made requests to be kept in the final version.
In-universe example in Midori Days: Seiji and his gang of delinquents get hired to play minor parts in a movie starring their favorite actor. During the final scene, the hero is mortally wounded. The boys are supposed to have a big mourning scene as he dies, but get a littletoointo the moment, and beat the crap out of the guys who "shot" him in revenge first. The director decides to keep it.
Because most anime is animated first and then dubbed to fit with ADR, in contrast to the Western practice of recording the voice first and then animating to the voice, there is a tendency for dialogue to be more ad-libbed.
When Norio Wakamoto was brought in to record the voice of Chiyo-chan's "father" in Azumanga Daioh, all of his scenes had already been scripted and animated. While he kept to the scripted lines, his delivery invariably ran longer than the animation (in one case, over a full minute longer). Rather than rerecord his lines, they reanimated the scenes to match them.
When Jan Valentine in the Hellsing OVA storms into the Council of Twelve's meeting room and faces a dozen weapons pointed in his direction, the original voice actor doesn't say anything before being shot. The English dub VA however ad-libbed "Oh, fuck me!" before the bullets started flying.
Lupin III: The Funimation dubs of the Lupin III films and specials feature a lot of this.
Lupin III: Crisis in Tokyo isn't a particularly funny movie in the native Japanese, but the dub had a ton of ad-libbing done by the actors (though not to the point of it being a Gag Dub), particularly Christopher Sabat, who voiced Jigen. It worked; it's one of the funniest Lupin movies ever released in the states.
In Lucky Star, Tsukasa's voice actress Kaori Fukuhara said the directors encouraged ad-libs. Her famous "barusamiko-su"note "Balsamic vinegar" — Tsukasa's Verbal Tic in the anime line was one of those moments.
According to legend, the names of the main characters (A-Ko, B-Ko, and C-Ko) in Project A-ko began because the creators couldn't think up good names for the characters, and started referring to them as A, B, and C during preproduction.
In Durarara!!, the kitty ears on Celty's helmet was originally a joke by Narita, but it was kept because the character designer really liked it.
In Afro Samurai, following the death of Brother 1, Ninja Ninja's "eulogy" ("Add one mo' body to the body toll, may God rest this po' bastard's soul") was impromptu on Samuel L. Jackson's part.
The Japanese version of Sonic X used this a few times. There is a scene in "Super Sonic Appears" where it looks like Sonic is going to get killed. His robot servants ask him if Sonic will survive to which Eggman was meant to have stayed silent (in thought of whether he could actually kill Sonic or not) however the voice actor jokingly said (in Japanese of course): "Of course! No one ever dies in animes!" The other voice actors decided to just go along with the joke causing the scene below to accidentally get created; the show's makers found the scene so humorous that they ended up keeping it in.
During the "Day of Sagittarius" episode, each of the SOS-dan members was commanding a space fleet, and each of them was shown in a bridge filled with Bridge Bunnies of their own imagining (It Makes Sense in Context). Mikuru's crew (a bunch of stuffed animals) were lifted from doodles that Aya Hirano (Haruhi's seiyuu and occasional Cloudcuckoolander) had drawn in the margin of her scripts.
When Kyon first meets the future version of Mikuru, she tries to prove who she is by showing him a star-shaped mole on her breast, and he stumbles back thinking "HUGE!" In the dub, Crispin Freeman borrowed a line from a fansub, saying "SUPERSIZE ME!" instead, and everyone thought it was hilarious, so it was kept.
According to Tiger & Bunny's scriptwriter, Nishida Masafumi, Origami Cyclone's habitual photobombing tendencies were something that he suggested to the producers as a joke. He was surprised that they agreed with the idea.
In Detective Conan, the character Wataru Takagi did not originally exist. He was a nameless character. In one scene, Inspector Megure asks his name and his voice actor (Wataru Takagi) responded with his name. Gosho Aoyama, the mangaka, kept it in.
On a whim, the artist of Busou Renkin decided to add a butterfly mask to the Big Bad of the first arc. This turns out to be fairly significant, as he is a recurring character, and the butterfly motif drives most of his personality.
In Inazuma Eleven GO, Hirofumi Nojima ad-libbed a Kiai as Kurumada, which sounded something like "shupoh!" It has since more or less become Kurumada's trademark exclamation.
In-universe: Negishi in Detroit Metal City (as Krauser) is invited to take part in an indie film by a director who turns out to be a Loony Fan and completely sold by the band's kayfabe. He also discovers that the female lead is an up-and-coming actress he greatly respects. The climax of the film turns out to be a sex scene the actress doesn't want to do, and Negishi, in order to protect her career, deliberately ruins it by ad-libbing in a physical assault on the male lead before the scene gets anywhere. The director considers the new direction the scene takes to be brilliant and leaves it in as a Twist Ending instead of re-shooting.
Dub example: John Ratzenberger's little soliloquy about the fat rich customer in Spirited Away.
In High School D×D, it was revealed in the afterword of Volume 12 that Ophis wasn't supposed to join the Occult Research Club group, though the author and his editor liked her design so much that he made her join anyway.
The soldier who is going to execute Shinji in End of Evangelion was always scripted to say it was Nothing Personal, but as originally directed the line would have come off as bloodthirsty (described by one of the voice directors as an "Eats Babies for breakfast" sort of voice). The voice actor instead read the line in a casual, almost joking, fashion. That version was used, as the result was even more chilling.
Minor example in the Lat Am dub of Dragon Ball Z. In this scene during the Cell Games arc:  in the original Vegeta only growls in anger at Cell's remark. However, Rene Garcia let out "He didn't just steal my cells, also my dialogues!!" as a joke during the recording. The studio found it so funny, they left it in.
In Noir, because her character Kirika is so laconic Houko Kuwashima was encouraged to do this, and wound up ad-libbing "there we go" at certain points where she found it appropriate.
Lyrical Nanoha has its entire franchise created due to one Throw It In moment - when redesigning Nanoha's magical girl costume after the "Lyrical Box" preview, legend has it that one took notice of the design and pointed out that it made her look like a Gundam. This essentially lead to a major upheaval and make the series as it is now.
The first of Takeshi Shudo's novelizations for the Pokémon anime states that Team Rocket's line said every time they're defeated, 「やな感じ」 "ya na kanji," lit. "bad feeling," was ad-libbed by their Japanese voice actors.