I speak Korean and I never had the possibility to hide it from my coworkers as it's written on my CV, so they knew I did when they hired me. I also ended up doing most of my interview in Korean.
My coworkers have all said they're relieved that I can speak Korean, and as of yet I wouldn't say I've been asked to do more work as a result - that said, I am not of Korean herritage. If you're a kyopo it can be different. With Korean history and society being the way it is, many Koreans don't really view ethnicity and nationality as separate things. This can cause problems for kyopos or people who look different from how Koreans expect people from their country to look. In the case of kyopos, some Koreans (I wouldn't want to completely generalise and say all because that's not the case) will reach a point where they will have to subconsciously group you into one category or other. They will view you as either a Korean who happened to be born in a different country, or they will group you as more or less someone "from that country".
If you come across Koreans who see you as the former - that is, a Korean who happened to be born elsewhere, then they may feel that you fit into their workplace hierarchy as a very low level Korean teacher who nevertheless can be used for their English language ability to do extra work, rather than as an important part of the English language education at the school, with a role that is unique to the native speaking teacher.
If I was in your position I wouldn't want to be lying about my language ability, for my sake and for the simple fact that at some point it's bound to come out anyway. I would tell people on my own terms. You don't have to tell everyone at the school or make a big announcement. Start by talking to your coteachers. Explain the situation. Do not get caught up in "showing off" your Korean speaking abilities (although you don't sound like the sort of person to do this anyway). Make sure you remain firm but polite and fair if people start to treat you differently from that point on in a negative way.
In my situation, if I ever do get people thinking I should be acting exactly like a Korean about a certian thing when I'm not, I point out that I've actually spent very little time here, that I grew up in and am part of a very different culture, and that language ability does not necessarily reflect who I am, or my cultural background. It's simply something I've learnt, a skill I've acquired, and I apologise if I'm not completely "Koreanised" about everything. I also ask for understanding of me on their part, and say I am and will always do my best to understand them a well.