I'm not sure how 'dyed looking' you want the hair to be but here are some that I've watched. I've found that a lot of Asian actresses dye their hair but it's usually a dark brown/red that looks somewhat natural so it's not as noticeable. You may be able to find some more unnatural colors in fantasy shows, of course a lot of that is just the actors wearings wigs.
Faith
Accidentally in Love
Full House Take 2; all three main characters actually have dyed hair
I'ts Okay That's Love
The Bride of Habaek
Strong Woman Do Bong Soon
Snakes and Earrings; the main lead also dyes her boyfriend's hair at one point
KBS Drama Special: Buzzcut Love