In The Queen's Gambit, Beth promised to pay him back $10 if she won her tournament but never sent the money or paid him a visit. She criticized the media for leaving him out of publication, leaving me to believe she cared for his part in her life, but why hadn’t she thought to return?
Miss Harmon had a busy life; full steam ahead. She went from orphan to hero. Its rags to riches theme is the reason we enjoy it so. Time was really of the essence for Beth. Go go go. Her mother, hesitant at first, soon realised her potential and encouraged her to go here there and everywhere, all whilst sampling the top booze everywhere she went. I don’t believe it ever occurred to Beth to pay the money the back, not because she didn’t love him, but because she had another two thousand things to think about. She had to study hard too, reading and ingesting as much as possible in order to succeed.
As for her insistence on mentioning his name in the article, yes, she cared for him. She did also feel awful (as explained in his lacking funeral). She wanted the world to know that this was the best janitor in the universe! One who loved her, taught her. Beth would hope internally that this would be satisfactory for Mr. Shaibel looking down on her in the world.
In addition to cmp's answer, I would add the less flattering observation that Beth is very, and problematically, self-absorbed for the majority of the story. In the various flashbacks to her (biological) mother we see this was essentially taught to her by her mother, who warned her of the men who would try to control her and their fragile egos and pride, and cautioned Beth that she was going to be alone and had to be strong and guarded. It's really only after Mr. Shaibel's death and the revelation that he was proudly following her career, and then later the revelation that Jolene had done the same and saw Beth as her sister and family, that opened her up to the idea that she had been terribly self-absorbed and neglectful of those important to her and helped her get where she was.
Her attempts to insure he was mentioned in the article was in part the culmination of this change of perspective, and opening herself up to the importance and relevance of others to herself and sharing that with others.
It also harkens back to prior articles written about her that we see her talk about. She is upset that they focus on her being a girl, paying little to no attention to her chess playing; "they don't even mention that I play the Sicilian", she remarks, or something to that effect. These are the sorts of things they would talk about with a male player, but with her it's about how she's this precocious little girl excelling at what boys and men do. Even at the tournament in Paris the reporters are still obsessed with interpreting her through the lens of femininity. The societal attitudes towards women during the mid-60's were incredibly sexist by present day standards. Her schoolmate that promptly got married and knocked up and became a (seemingly unhappy) housewife was the one that was doing things right, as far as societal norms and expectations were concerned. Beth has a strong desire to be treated in the same way as the males, to be viewed as a "chess player" and not "a female (chess player)". Hence her excitation at seeing Borgov ditch his ongoing match just to see how hers had ended, and how disarmed she is by the praise she receives from the guy with the big hair (I forget his name) and ultimately Borgov. These reactions arise at least in part from her delight and surprise at finally achieving this equality. It was praise not predicated upon her being a female, but simply on her excellence in chess. That the press at this tournament end up asking her such "chess" questions, such as if it was true that she started playing at 2 years old, were also a signal that she had finally earned respect enough to be valued independently of her gender, and that talk of her style and history with chess were now important enough to talk about and be printed.
Anger and fear. After the episode with the pills, they refused to let her go to the basement to play chess with Mr. Sheibel. This obviously hurt her deeply, and she begged him to help her out and he didn't. She grew to hate the place. Show also had an affection for him... A feeling which causes fear and anxiety in her. She often pushed everyone she likes out of her life so she could stay emotionally safe.