It is safe to drink tap water in Korea but as you may see, most of restaurants will have filtered water available. In any place in the world, the practice of boiling tap water before drinking is a good idea. Nonetheless, it's odd to find such question for Korea...

and see a post by someone claiming that they got sick after drinking water in a restaurant in Korea - how do you know if the sickness is caused by water? There are many confounding factors.