Crash landing on you Captain Ri pretended she was his fiancée to literally save her life worst case scenario/prison best case scenario.

I think Business Proposal is one of the more obvious choices for this. The leads are already in a fake relationship for the benefit of grandpa, but Tae-moo turns it on for the benefit of her crappy "friends" too (when she's alter-ego-less) and I loved pretty much everything about this part of the drama.

I think Her Private Life also fits pretty well. In this case, the ML is saving her from truly fanatic fans and they plan the whole fake relationship out, but it stems from him just yelling that she was his girlfriend at a random car that they thought (incorrectly) was attacking her.

Search WWW is one of my favorite instances of this trope. Hyeon needs to pretend she's in a relationship in front of Ta-mi, so she calls the only person she can think of, which just so happens to be a man who is actually in love with her. Ji-hwan shows up, they get their story straight before heading into the bar, and then he proceeds to tell their actual story to Ta-mi. If you didn't love Lee Jae-uk before, this should do it!

Honorable Mentions: These are the ones that might fit, if you squint just a little.

The King's Affection: I just love Rowoon's bisexual self in this drama. It's just a scene, but I loved it: the crown prince, who is actually a woman, is disguised as a woman to get onto a ship. Ji-un, who thinks she's a man, pretends to be her husband to shield her from the prying eyes of those searching for the crown prince. It's got to be one of the most gender-bending episodes of any drama and I didn't know whether to hold my breath or laugh.

Also Because This Is Our First Life is literally all about a fake relationship, but the scene where Sehee comes to Ji-ho's rescue is one of my favorites. It's not to save her from embarrassment, it's to actually save her from a creep, but it was quite a swoony scene nonetheless.

Her Private Life was memorable with this one too.
And Because This Life is Our First.
You Are My Spring. Literally saved her life.
Business Proposal.

She Would Never Know The FL broke up with her boyfriend who didn't really take it well. The ML helped her but the best part is she's the one who took the decicion (you need to watch it to understand :p)

Another Miss Oh She was in a lot of embarrassing or kinda dangerous situations and he helped her a lot.

Suspicious Partner Her boyfriend cheated on her and the ML helped her much to her surprise :D

My Husband Mr Oh is one good drama. Kim Kang-woo and UEE are *chef's kiss*.

Does love in contract count🤔

I will never stop recommending That Fool.

Train, though it is only for a few seconds. But I love that show.

Deserving of the Name, although it was the reverse of rescuing... 😂

Dal-ja’s Spring is an oldie but (IMO) a goodie with this exact trope at the end of the first ep I believe? A classic contract relationship drama with the male lead as a for-hire fake boyfriend. Some parts haven’t aged well but for the most part it holds up pretty well (esp the important parts: the eventual “real” romance and the leads’ characterizations were quite ahead of their time imo).

Business Proposal of course is super winning for this trope as well!!

Romance is a Bonus Book, when Ji Seo-jun (Wi Ha-joon) rescues a barefoot Kang Dan-yi (Lee Na-young) from an obnoxious drunk, and then crouches down Prince Charming style at her feet to put (her own) shoes on them. He'd seen her losing one under a bus that morning and rescued it, afterwards finding its discarded twin where she'd dumped it in a bin.