Just what the title says. I've heard that it's because we don't supplicate Hashem on shabbat, but this seems a little simplistic. What are the issues here?
The Levhush Siman 584 Seif 1 writes that Avinu Malkeinu was authored by Rabbi Akiva to mirror the Berachos of the weekday Shemona Esrei,
Avinu Malkeinu Choneinu V'Aneinu = Chonen Ha'Daas
Hachzireinu B'Teshuva = Horotze B'Teshuva
Selach Umichal = Selach Lanu
Kosveinu B'Sefer Geula = Goel Yisroel
Shelach Refua = Refainu
Chadesh Aleinu = Borech Aleinu
Horeim Keren = Teka B'Shofar
Bateil M'Aleinu = Hoshiva, V'hoser Mimenu Yogon V'Anacha
Kalei Kol Tzar = Shoveir Oivim
Mechok B'Rachamecho HaRabim = Yehemu Rachamecha
Horeim Keren Meshichecha = V'Karno Torim
Hatzmach Lonu = Matzmiach Keren Yeshua
Shema Koleinu = Shomeya Tefila
and it also includes requests for the Tzorchei Rabim.
The Levush finishes that when Rosh HaShana and Yom Kippur come out on a weekday, which if it was not Rosh HaShana we would of Davened the full Shemona Esrei then we say Avinu Malkeinu which mirrors the Shemona Esrei. However when it comes out on Shabbos where we Daven a different Shemona Esrei we do not say Avinu Malkeinu which mirrors the weekday Shemona Esrei.
Avinu Malkeinu is a prayer in which we beseech Hashem for healing, sustenance and other things that we need. On Shabbos we do not pray directly for our needs. This is because referring to the material things that we lack can cause us distress, something that is inconsistent with the Shabbos atmosphere of pleasure and bliss.
NB: Sepharadim (at least the North Africans--Moroccans, Tunisians, Algerians, Libyans--with whom I daven) do say Avinu Malkenu on Shabbat Shuva.