Providing paid accommodation without registering at the town office is illegal, if it is done more than 7 days a month. However, as @BanzaiTokyo said, this is the problem for the host, not you.
A few tips to remember while staying, are:
- Rules for dumping garbage are quite strict in Japan. Make sure that you ask the host how to deal with trash.
- You will most likely be staying in a residential environment. Please do not make too much noise.
- Japanese apartments are low on insulation. If you are traveling in winter, do check whether heating is available.
- Just in case, ask the host and find out what to do and where to evacuate to, if there is an earthquake.