No, Koreans do not eat their pet dogs. Many Korean people keep dogs as pets and members of the family, just like in Western nations.
There has historically been a specific breed of dog raised in East Asia for food. It is not, as Internet rumors have spread around, the St. Bernard. However, since South Korea outlawed dog meat in the 1980s around the Seoul Olympic games, there has been virtually no regulation of the technically illegal dog meat industry in Korea (the government turns a blind eye). And this has led to allegations of horrible conditions at dog farms and slaughterhouses.

