A very common question is, "Do I have to speak Korean in order to teach English in Korea." The answer is: That's not the right question to ask. The question you should ask is, "Do I have to speak English in order to live in Korea. Answering that question will help you determine how much Korean, if any, you should learn.
As far as the job goes, you do not need to know Korean in order to teach English. Of course, it doesn't hurt. In fact, it can be quite helpful at times, "like when your students have no idea what you're talking about and you just want to convey a simple message". But it is not necessary by any means.
The real question has to do with what you want to do outside the classroom. If you live in Korea, then there is a certain amount of interaction with non-English-speaking-Korean-folk that will be unavoidable. Take, for example, ordering food at McDonald's (or ordering food just about anywhere for that matter), or trying to find the toilet paper at E Mart, or trying to pay your bills at the bank. In Korea, despite the governments best efforts and intentions, people speak Korean. So unless you're really good with body language, a little bit of Korean will take you a long way, and make you life a lot easier, and more enjoyable.
For example, it's a lot easier (and more polite) to say, "이것 주세요," when you're order food than it is to just point hope that the waiter gets the message. In the taxi is another place where knowing a little of Korean will really save the day. Because good luck finding a taxi driver who knows how to say anything other than "hello," "left," "right," and "straight". And most of them probably couldn't even get that far. So it's not like you have to learn how to "speak Korean". Just a few key words and phrases can help a whole lot.
A great place to start is survivalphrases.com. You can download free podcasts, as well as other useful info about not only Korean, but a wide variety of languages. Another great resource is youtube.com/busyatomdotcom. He makes great videos about learning basic Korean.
Learning Korean is actually a lot of fun, and it can earn you a lot of respect at work if you let people know what you're doing. It can also make you life more enjoyable and easier outside of work. It's also a great way to make friends and meet new people. So give it a try!