Happy New Year in French is Bonne Annee. Furthermore here are some more translations of Happy New Year in other languages:
Afgani - Saale Nao Mubbarak
Afrikaans - Gelukkige nuwe jaar
Albanian - Gezuar Vitin e Ri
Armenian - Snorhavor Nor Tari
Arabic - Kul 'am wa antum bikhair
Assyrian - Sheta Brikhta
Azeri - Yeni Iliniz Mubarek!
More Happy New Year in different languages here.