Romantic Doctor Teacher Kim 1 and 2. There's romance between the young doctor mentees, but not with their mentor.

I found the relationship between a mentor and a mentee in Be Melodramatic very refreshing.

Obviously Jung Myung-seok to Woo Young-woo in Extraordinary Attorney. Best mentor ever. I still haven't forgiven the writer for making him ill.

But also:

Mr Sunshine has a great one. Once her grandfather gives up on Go Ae-shin staying out of the conflict, he enlists Jang Seung-goo to train her to be a sniper. He totally doesn't believe she's going to stick with it at first, but they gain each others' respect over time, and have some really beautiful (and heartbreaking) scenes together.

One that I really really loved (maybe my favorite?) was in the Crowned Clown (I know that some folks hate this drama with a fiery passion, but I am not one of them). The relationship between Haksan and Ha-seon was what made the show for me. It was initially combative, as Haksan strong-armed Ha-seon into doing this thing that he had no desire to do, but as Ha-seon proved again and again that he was up to the task, Haksan started to respect him more and more until they ended up with something closer to a friendship or a father-son relationship. Haksan is already such a fantastic, complex, and ambiguous character that it makes their relationship progression really powerful.

I also think Brian from Search WWW is a really great mentor. He pretty much can't retire because his employees keep finding him and asking him for advice and he is a comforting fatherly presence in the show.

And finally, no rec list would be complete (at least from me haha) without a Live Up to Your Name recommendation. ;) It's not a huge part, but Heo Im is a mentor to Dong Mak-gae who also wants to be a doctor to the point that she dresses as a man in Joseon so that she can assist him. Normally in a drama there would be jealousy when the FL showed up, but I liked this relationship because they never went there, and maintained their totally platonic interactions even though Heo Im totally knew the whole time that Mak-gae was a woman.

Healer : Yoo Ji-Tae was the mentor of Park Min-Young, they had a childhood connection but there wasn't any romance between them.

Hot Stove League : Even if Namkoong Min knows nothing in baseball, I think he was still a mentor to Park Eun-Bin and there wasn't any romance between them.

Tomorrow : Rowoon was only the mentee and whipping boy of Kim Hee-Sun.

My rec would be to avoid My Roomie Gumiho at all cost.

Rookie historian Goo Haeryeong was the first thing that popped into my mind. How the deputy chief historian (Min if I remember right) guides her, reprimands her but still stands up for her when she makes a mistake, learns from her without being petty about it.

Yoo Ji Tae’s and Yoo Hwa Young’s characters in Mad Dog.

It's a bit disturbing that romantic mentoring relationships seem to be the default kind in K-drama. Anyway, I must fly the flag for a great female mentor, Nobody Know's Cha Young-jin. Not only is she the steady, responsible adult in Eun-ho's life, she's also a good boss and senior to the two youngsters on her team.

Misaeng comes to mind. Of course, that is not a male-female pairing, but great mentor relationship, nonetheless.

My Mister might count, but only if you can stand one-sided semi-romantic interest.

Racket boys! The coaches were great (in the end). ^^

Chief Kim and his boss in the accounting department?

Lots of parent/grandparent roles (e.g in Start-up)

School 2013? I don’t remember there being any romance.

I think one semi-overlooked Mentor-Mentee relationship is in Forest of Secrets... our stone faced prosecutor gave some great advice to not only the sleazy waffling bad guy prosecutor but also the newer female prosecutor. I don't remember their names ahaha

My name. I really liked ji woo and Choi mujin there.Their chemistry was something else.

Does Special Labor Inspector Jo count? Lmao
Also Hot Stove League.

Would Lookout count? I mean, it was both a chase and mentorship for vigilante style justice.

Never give up. No obvious romance. Lovely show overall.

I propose the wonderful NAVILLERA (2021) (Netflix US).
I am not sure it can be considered a pure mentor/mentee drama when it turned a teacher/student drama on its head with ‘youngster’ Chae-rok (Song Kang) forced to give ballet instruction to ‘senior’ Deok-chul (Park In-hwan). The ending of episode 8 was just stunning. One of my favorite dramas of 2021.

"Live" Lee kang soo and his mentor were brilliant from hate to love
Even FL and the woman police officer (Lee kang soo's mentor ex wife) were so good together but their relationship had just a little screen time.

Argon (2017). Kim Baekjin as an anchor and reporter gave a great deal of trust to the newly joined team, Lee Yeonhwa. Though he was flawed, he showed her the good way and didn't let her repeat his mistakes.