I am not going to give a definitive answer on this. I will instead present some facts and discuss my opinion on the matter and let you (the reader) make up your own mind.
Some people say “when in Rome” meaning you should do what the locals do. If you ask around the general consensus in Korea is the water is not safe to drink, but no one can tell you why. I find this amusing (but I am not surprised) because there is also a widely held belief among Koreans (not all Koreans) that going to sleep in a room with a fan turned on is DANGEROUS!!! They call it fan death. So if you believe in blindly following the “when in Rome” idiom then I suggest turning your fan off when you sleep at night.
Historically it was not safe to drink the water (as with any developing country), but a lot has changed in Korea. When you consider that only 25 years ago many Koreans still lived in huts and cooked over a fire it is not surprising that there are many (older) people that still think it is unsafe to drink the water. Culturally in Korea people (usually) don’t question the status quo. By this I mean they accept things are the way they are and don’t try to change them. In this case they have grown up with the belief that it is unsafe to drink the water, they have no desire to question that belief, so they continue with the current practices of buying bottled water, boiling water or using a water purifier.
Koreans can also be hypochondriacs when it comes to health. Whenever I have a runny nose or a cough, my co-teachers say “you should go to the hospital”. I have to insist I am fine and will be OK in a few days.
There are people who post on the internet and claim that the water isn’t safe to drink and site a personal experience where they have gotten sick after drinking the water. I am sure they got sick but I can’t help but question if their sickness was caused by something in the water. The stories of westerners who drink the water without any ill effects far outweigh people who believe they have gotten sick from the water.
The Korean Ministry of Environment website categorically states that water must pass strict tests before it is supplied to each home and that is safe to drink 365 days a year. The website doesn’t go into specific detail of what each test is, just that (depending on which city it is) they carry out between 120 and 164 tests.
There is also the conspiracy theorist in me that says “maybe certain companies are feeding people misinformation to maintain the strong bottled water and water purifier market”.
Conspiracy theories aside, there is one consideration for why you should drink filtered or bottled water and that is the smell and taste. Personally, the smell and taste doesn't bother me but it does bother some people, however a funny smell does not mean the water is unsafe to drink. In Australia the water sometimes smells funny during a drought because the water company has to put more chemicals in the water to make sure it is safe to drink.
At the end of the day bottled water is cheap (less than 400won or $0.40 for a 500mL bottle at the supermarket) and often even free in hotel rooms. Restaurants, train stations, workplaces, hotels, schools and many Korean homes also have water purifiers which are free to use. If you want to play it safe, you won't have any trouble finding filtered water to drink.