The South Korea is going to overthrow its government. This is evidenced by the large group of protesters who have gone on the roads to protest against the government of South Korea. I therefore believe that they will manage to overthrow the government since the citizens are so powerful people in the government.
It is highly unlikely that South Korea will overthrow its government. South Korea is a highly industrialized, Westernized nation, and coups simply do not occur in those nations. It is my belief that South Korea will have a peaceful, relatively uneventful transfer of power, similarly to the way there will be a peaceful transfer of power to Donald Trump in the United States.
There is probably nothing to worry about. South Korea may just oust one politician for another. The government should be able to survive this transition. However there is a small possibility that it will be overthrown. There is really no telling with governments of other countries. What the people want is most important.
South Korea lives under constant threat of tyranny and war from North Korea. The South Korean government's current policies, though disagreeable, will not lead to a coup de'tat simply because the South Korean people know the alternative. They cannot afford to have a weakened and unstable government for any amount of time because of North Korea's military intentions.