Historically Seoul has had a number of issues with drinking water.  Most recently the issue wasn't the water itself, but the pipes that delivered it, which many people claimed introduced contaminates.  I can still remember being told by someone I was working with there about 10 years ago that the water was safe to drink in my hotel, but not in their office block located a few blocks away - I'm still not sure exactly how he came to that conclusion!

Over recent years the local government-run water supplier has made significant efforts to replace all of the "old" pipes, and has taken additional measures around treatment of water.

The water is considered safe and drinkable under UN criteria, and although many locals still do not trust it, there's little doubt that it's as safe as tap water in any other major first-world city.

There's numerous more information available on [The Office of Waterworks, Seoul Metropolitan Government][1] website.

Note that all of the above comments relate to the water in Seoul.  If you're heading outside of the main metropolitan area then things could be very different and I would suggest only drinking bottled water in those areas.
  [1]: http://arisu.seoul.go.kr/sudo_eng/index.jsp