The 2014 miniseries "Secret Love" ("시크릿 러브"), consisting of 5 romantic stories featuring the members of the KPop group KARA, had an episode filmed entirely on Jeju Island entitled "Have You Ever Had Coffee With An Angel?" Ji Chang Wook and KARA's Park Gyu Ri were the leads.

Extreme amounts of fluffyness (and a sprinkling of angst), but was worth watching for the gorgeous scenery whilst the characters (who worked for a travel agency, mind you) travelled around showing us the hot spots of Jeju-do.

The entire series is out there on the usual dodgy illegal streaming sites (which I refuse to link here). 🤬

And RIP Goo Hara. 😭🙏

My Girl comes to mind - the beginning was in Jeju. Gives you a glimpse of early Lee Joon-gi and Lee Dong-wook and is also a fun rom-com (for the most part).

I know you've asked about dramas, but what came to my mind was actually Hyori's Bed and Breakfast 1. I love the first season, and you can appreciate what Jeju has to offer a great deal. I haven't finished season 2 yet, though. Season 1 was also really special because IU was their helper and she was a sight to behold. Very creative and artistic.

As always, I am here to say: Kim Sam Soon!
Instead of being at the beach, the Jeju parts were on Hallasan, the big mountain we see in the background on all the other dramas 😉
Hilarious, touching, memorable and made me a lifelong fan of Choco Pies.

Oh wait! Also there are scenes involving a hotel opening ceremony (and a knock-down, drag-out fight), some lovely touristing by the 2nd couple (learned to say “do I look like a pushover? in Korean), and a glimpse of the shore when the “hero” dumps the heroine at the side of the road. Oh my gosh. So many funny things happen in this drama.

In two of my first dramas the characters relocated to Jejo at some point in the story. My Girl, and Dae Jang Geum.

I liked HIGH CLASS. Jeju locations were used as reflective places for the characters. It also brought home the isolation of Jo Yeo Jeong's character against the status, power and behavior of the rich school mothers.

Boys over Flowers
Secret Garden
The Legend of the Blue Sea
Something in the Rain
Lie to Me
Scent of a Woman
Fated to Love You
Crash Landing on You

The final episode of Sh**ting Stars is set mostly on Jeju.

There are a lot of dramas with some scenes on Jeju Island (Secret Gardem, The Legend of Blue Sea, My Lovely Sam-Soon, Something in the Rain, Love Alarm 2, etc.) but the whole drama, it's rarer.

Dramas that take place on a island :

Top Star Yoo-Baek was filmed on Daemo Island, the scenery was beautiful.

Hospital Ship had a beautiful scenery too, taking place on small islands.

There's a lovely little web drama from a few years back with So Ji-sub and Kim Ji-won called One Sunny Day which takes place on Jeju. It's an adorable little story about two strangers who meet by chance. I believe it was made in part to highlight Jeju as a tourist attraction, so the island actually plays a large role in the story itself and its natural beauty is highlighted throughout. It also has a killer soundtrack which intoduced me to Ggotjam Project. It's one of my favorite web dramas and and easy binge.

On the Way to the Airport has a few Jeju episodes.

Several dramas featured the Seaes Resort on Jeju which some Beanies have actually visited! It was used in
The Snow Queen
Goong (Prince’s bday party)
Secret Garden
And probably many others.

koreandramaland.com has lots of listings of dramas and their specific Jeju locations (hi, @michh!)

When you start thinking about it, you almost wonder which dramas don’t have scenes in Jeju.

There’s an old one called Loving You (Lee Dong Wook is the 2ML, it’s so old) set on Jeju where the family owns a clementine farm.

Other than what already is mentioned:
Warm and cozy
Jewel in the palace (part of thr drama)
A part of Marriage blue(movie)

Check out the event

It is a minidrama that completely happens on Jeju osland

Tamra Island from 2009 was set in Jeju! Honestly haven't watched since, so it might be widely outdated, but I remember thinking it was fun back then.

TAMRA THE ISLAND literally refers to Jejudo- it was an older name for the island.

My first Jeju do drama was Swallow the Sun 2009, with Ji Sung. The island captured my imagination, especially the sea and the Haenyeo. Ever since then, I'll watch anything that is set there, and one day I'll go. So I guess the product placement works.

I’m not sure if it was filmed on Jeju or not but the first two eps of Pinocchio were filmed on an absolutely gorgeous island.

This is not a drama recommendation but a movie. It's Architecture 101 starring Suzy/Lee Je Hoon (as the younger version of FL & ML) and Han Ga In/Uhm Tae Woong. ML is an architect and FL wanted him to re-design her late father's house in her hometown - which is in Jeju!