Yes, Every Foreign Residents should report the change of address within 14 days of the moving date, or you will get fined.
You can either report your change of residence to the immigration office or district (gu) office for that. these are the process of how to report the change of residence for both ways below
[Report to immigration office]
1. Make an Online Application
please visit this link: https://www.hikorea.go.kr/pt/CvlapplInfoPageR_en.pt?locale=en&
2. Click for the Notification of Change of residence. (e-Application)
then, the next page will tell you the required documents and process.
3. Once you have made an appointment, you need to bring the application form, passport, ARC, housing lease contract. if your accommodation is under another person's name, you have to submit "거주숙소제공사실확인서" along with the other documents.
[Report to district(gu) office]
1. you can visit your district (gu) office 민원센터 (Minwon center). the documents are the same, Passport, ARC and a housing lease contract. the application form is "Alteration of residence / 체류지 변경신고" and you can find this form at the office.
Again, if your accommodation is under another person's name, you have to submit "거주숙소제공사실확인서" along with the other documents. they may request provider's ID card copy as well.