Concerning home country taxes, detailed information can be found in the General Information Handbook, which is accessible from the JET Programme official website.
The United States, the Netherlands and Ireland have a tax treaty with Japan which makes JETs from these countries exempt from the Inhabitant’s tax for their first two years in Japan.
However, if you are not from one of these countries or are a 3rd year JET or above, the bill for the inhabitant’s tax will be sent to you around June. Because you are leaving, you will not be able to make quarterly payments (without adding a number of difficult, unnecessary steps involving your tax representative and many possible bank transfers), so please be aware that depending on how your CO handles this tax (please ask the person in the office who is responsible for your financial affairs) you may have to pay up to 170,000 yen at once in June. But don’t fret! This will be reimbursed by your CO in July, (presumably) before you leave.