All persons must pay taxes on any income they receive in Iceland, regardless of nature. The taxation system in Iceland is a PAYE system (Pay-As-You-Earn). Employers are required to deduct taxes from all salaries paid out to employees. Generally employees apply for a tax credit, see https://www.rsk.is/english/individuals/personal-tax-credit/ for more details. Individuals must have an Icelandic ID number and non-EEA/EFTA citizens must also provide a copy of their work permit in order to apply.
All students who have received income in Iceland, including salaries, grants, or other forms of remuneration, must file a tax return to the Directorate of Internal Revenue (RSK) annually and no later than one week before leaving the country. See https://www.rsk.is/english/individuals/tax-return/ or contact the RSK directly for more detailed information. Exemptions may apply, for more details see https://www.rsk.is/english/individuals/exemption/.