A week ago I listened to an episode of Disrupting Japan podcast where its host Tim Romero spope at length about difficulties AirBnB is facing in Japan and the possible developments of the situation. It was quite interesting and shows how business in Japan is different from elsewhere in the world.

But it is not important if you are just visiting Japan. If I understand it correctly, the only breaking of the rules happens on the side of the home owners. So I would guess that you don't risk anything if you book through AirBnB.

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:

1. Rules for dumping garbage are quite strict in Japan. Make sure that you ask the host how to deal with trash.
2. You will most likely be staying in a residential environment. Please do not make too much noise.
3. Japanese apartments are low on insulation. If you are traveling in winter, do check whether heating is available.
4. Just in case, ask the host and find out what to do and where to evacuate to, if there is an earthquake.