The Avox Girl
In the book, Katniss learns about Avoxes after she recognizes one to be a woman that she had seen before. The encounter happened in the forest near her District, where she watched the woman get chased down and taken away by a Capitol hover craft.
The Tribute Inspired Muttations
In the first book, Katniss realizes with horror that each beast mutation that is sent to them at the end of the game closely resembles the fallen tributes. The idea that the beasts could actually have been made out of the bodies of the dead tributes comes close to pushing Katniss over the edge.
Katniss' Suffering In The Games
In the movie, Katniss finds water quickly after entering the games, but in the book, she goes days suffering from dehydration. The pages describe her struggling to pass urine, and ultimately falling to the ground ready to die before she accidentally stumbles upon drinkable water.
This showed the very real probability of dying in the games, which was important for Hunger Games audiences to know. However, the book had days upon days to allow the Games to play out, while the movie had to condense the events greatly. Therefore, Katniss' suffering at the hands of other players or the Gamemakers took priority.
( @thgmeta : just say that you want a PG 13 rating movie)
Peeta's Leg
In Hunger Games Peeta's leg must be amputated after the first games. For the rest of the series, he wears a prosthetic. The lost leg represented the suffering that Peeta had faced in the Games, and also increased the threat that he faced re-entering the arena in Catching Fire.
Having an actor appear to have a prosthetic leg would be more difficult than audiences might think. It would have likely required special effects with CGI, which would have, of course, cost more money. In the end, this would have been a lot of work for what was, comparatively, a rather small detail. In the end, the small change in The Hunger Games movie was best.
(@thgmeta : What?!)
The Mayor's Family
In The Hunger Games book, Katniss is given her mockingjay pin by the mayor's daughter, Madge Undersee. She later learned that the pin had belonged to Madge's aunt, who had been a District 12 tribute, and Katniss' mother's best friend. The gesture solidified their friendship, but Madge would, unfortunately, die when District 12 was bombed in Catching Fire.
Katniss Singing
Katniss' father was described to have had a beautiful voice, which, in the books, he passed on to Katniss. With the responsibility of her family's survival on her shoulders, however, Katniss saw little value in music. When she finally did sing, the people around her were in awe and even the birds were said to stop and listen. Katniss' singing even relates to Snow's story in The Ballad of Songbirds and Snakes.
Katniss does sing in the movies, but the films chose to remove scenes that focused specifically on her singing ability. This likely came down to actress Jennifer Lawrence's own singing skills. Lawrence played a phenomenal Katniss and sang her war cry song well, but her singing voice didn't have the magic the books described, so this detail had to be scrapped.
( @thgmeta : hmm, you can dubbing with another voice actor? Or auto-tune )
The Lake In The Woods
In Catching Fire, Katniss visits a lake in the District 12 forest that she and her father would visit when she was little. She had learned to swim there, and the father and daughter duo would harvest katniss roots, her namesake, from the lakebed.
Haymitch's History
In Catching Fire, Katniss and Peeta end up watching a recording of Haymitch's games and learn a lot about his character. Similar to them, Haymitch had made the Gamemakers look foolish when he won, and the result was that he arrived home to find that anyone he had cared about had been executed.
In the movie, Haymitch's character is considerably watered down, likely in an attempt to make him more likable to audiences. His drunken tendencies often made him mean and messy, with a special dislike for Katniss. The revelation of his past explained his behavior and granted Katniss some empathy for him, but since this version of Haymitch's personality was removed, the scene describing his past would have had little impact.
( @thgmeta : don't worry there is a well fanmade video of his games)
Katniss Killing A Capitol Citizen
Katniss' brutality is also greatly watered down in the Hunger Games movies. In the books, she could kill without thinking, only later lamenting at how easily that Capitol had turned her into their monster. In Mockingjay, when Katniss and her team are trying to make it through the Capitol, Katniss shoots down a civilian to ensure that they would not draw attention to her group.
Katniss often put survival above morals, which made her a difficult person for people to like. However, her inner dialogue in the books allowed audiences to empathize with her choices. Movies lack the ability to always show what is going on inside a character's head, so Katniss' brutal actions had to be removed to ensure that audiences could sympathize with her.
Katniss Becoming The Girl On Fire
In Mockingjay Part 2, Prim is killed when hidden bombs in the Capitol square detonate. The explosion sends Katniss flying backward, and she loses consciousness. However, the scene in the book describes both Katniss and Prim going up in flames. While Katniss became the literal "girl on fire" that the Capitol had dubbed her, she was forced to watch her sister burn alive.
This terrible and heartbreaking scene may have been too brutal to include in the movie. Katniss barely survived, with a majority of her body scarred forever. However, she was numb to all but the loss of her sister, whom she watched die in the most terrible way possible. While the impact was stronger in the book scene, including it could have potentially changed the rating of the film.
Not every book scene can make it into its film adaptation, and some skipped Hunger Games scenes just simply couldn't have worked.