The last perek of Sanhedrin lists some people who have no share in Olam Haba. Other people who forfeit their share are discussed in the answers to this question: No Olam Habah for these?
My question is: What happens to those people who don't have a portion in Olam Haba? Do they stay in Gehenom? Disappear? Something else?
Me'iri explains that they go directly to Gehenom.
According the Rambam Ramban, after death you either go to Gan Eden or Gehenom. At the time of the final judgment it is decided if you go to Olam Haba, these people do not stop for judgment but remain in Gehenom so as not to increase their punishment.
Yad Ramah's explanation is similar to the Ramban but says the reason they do not stand in judgment is that they already had their punishment here and should not be further punished.
The Rambam explains, in line with his explanation of what Kares is, that they are cut off from their own spiritual component, and they remain physical. When the physical world comes to an end (as it will, in the Rambam's world), these people will end with it. They have no connection to spirituality, as they have chosen to involve themselves and invest in physical pleasures and endeavors, and therefore they are cut off from spiritual existence. When Olam Haba, which is a purely spiritual/intellectual experience according to the Rambam, comes, they will have no way to experience it and will cease to exist.
Rambam Teshuva 8:5
הנקמה שאין נקמה גדולה ממנה--שתיכרת הנפש ולא תזכה לאותן החיים, שנאמר "היכרת תיכרת הנפש ההיא, עוונה בה" (במדבר טו,לא). וזה האובדן הוא שקוראין אותו הנביאים דרך משל, באר שחת ואבדון ותופתה ועלוקה; וכל לשון כליה והשחתה קוראין לו, לפי שהיא הכליה שאין אחריה תקומה לעולם וההפסד שאינו חוזר לעולם
Rambam Intro to Perek Chelek (Shilat translation):
הנה כל מי שדבק בתענוגים הגופניים ומאס באמת ובחר בשקר, נכרת מאותה מעלה וישאר חמר נכרת בלבד
Those who do not have a portion in Olam Haba cease to exist.
