You lived in Korea, so you speak Korean? How much of a language do you need to know to teach English to the learners?

As a traveller I do think it shows a great amount of disrespect not to learn at least some of the language when you go to a different area. Even if I was spending less than a week in a country, I would always learn: please thank you, where is the, and no. I found it earned me some respect and overall it is just the polite thing to do.

So yes, I picked up some Korean (now I remember very little) and when I lived in the Netherlands I learned some Dutch.

As a teacher however I do not feel you need to know a learner's language in order to teach them. I do think you need to understand their language linguistically to anticipate problems. For example, I did a poetry unit in Korea. I did not know that Korean's do not rhyme. The best translation of the word rhyme is poem, but they aren't familiar with the concept in their L1. If I had known that I would have planned more time on the actual skill of rhyming (which I ended up doing anyways)

When I took my CELTA  we had a copy of Swan's Learner English that I found invaluable for understanding why a student may struggle with a lesson.

I do agree with others who have said that it is very helpful to teachers if they have learned a second or third language as a student. This gives you tons of skills that later help you as a teacher (and you may be able to snag some activities and games!)

So basically no I don't think that English teachers need to be multi-lingual. I do think it helps, but I think most people would benefit from being multi-lingual, not just teachers.