Can I purchase a house in Korea and use OHA to pay off the mortgage?

I don't know how it is in Korea, but in Japan the housing office had to inspect and bless off on standards of living in an apartment before being eligible for OHA.
Presumably, your house would have to meet these standards and be inspected by post housing prior to being eligible for OHA.

Buying a house in Korea is a lot different then buying the house in the states. If the house is in your family members name and you are authorized to live off post, then there is a possibility. Even if you have family on your orders, does not mean that the will even allow you to live off post. When I first got to Camp Humphreys, they checked the availability of on post family housing, under the assumption that if on post housing was available, then off post housing was not authorized. Luckily, no housing was available, so I moved off post.

If you have an approved command sponsorship and on-post housing is available, they will not provide you OHA.