According to the Ministry of Justice website, starting April 1, 2010, anyone extending their Japanese visa or changing their residency status will be required to show a valid health insurance card/booklet. The relevant portions are below.
While technically the rules have already required registration in the national health system (technically it is a duty of almost all residents, citizens included) there has not previously been any penalty for non-registration, although I hear a history of insurance non-registration it may cause problems when applying for permanent residency or citizenship. Note that although in principle residents are supposed to be registered in both the national health insurance and the pension scheme, these regulations refer only to the health system, and it does not seem that non-registration in the pension scheme will have any effect on visa renewals.
I have met an awful lot of foreign residents of Japan over the years who have never bothered to register for either public social insurance program and have never had any trouble over it, but this will not be the case in the future. Any Japan resident foreigners not registered in the national health insurance who are thinking of extending their stay past their current visa term had better look into registration ASAP.