Does stage 3 rectal cancer cause serious longterm effects on continence?

Depends. On how low in the rectum the cancer is and what modes of treatment are proposed or undertaken. If it is low it may indeed affect continence and a permanent colostomy may be indicated.
Not often. The cancer itself can invade the muscles of continence for really low rectal cancers that are near the anus. Many rectal cancers do not invade the sphincters. Surgery and the radiation that is recommended for stage 3 rectal cancer can contribute to incontinence but it is very difficult to predict and it is by no means a guarantee that incontinence will occur.