Yes, it is safe to drink tap water in Seoul but it depends who you ask.
In the past, the tap water in Seoul has been considered unsafe to drink because of contaminates caused by old pipes that the water was running through.
Since then, pipes have been replaced, water quality has improved, and according to the UN and the government, the tap water in Seoul is considered safe to drink. The tap water has passed all necessary tests multiple times to ensure its safety.
As of 2016, the tap water in Seoul, known as Arisu is considered to be among the safest water in the world to drink. The water goes through 163 water quality tests, more than most developed cities.
Even with all the changes, many residents of Seoul still choose to drink bottled water as they still have worries about the quality of water pipes. It takes time to change from something you are so use to doing your whole life.
If you are visiting Seoul, you should have no worries about the safety of the tap water. In due time, the residents of Seoul will most likely also make the switch over from bottled water to tap water.