Costa, 36, became the first female coach of a professional men's club in the top two tiers of any European league when she was appointed last month.
No explanation has been given for the Portuguese women's decision not to assume the post, which the club have described as "sudden and surprising".
Claude Michy, the Clermont president, said in a statement: "Helena Costa has decided not to honour her commitments in relation to Clermont Foot 63 . . . This decision was sudden and surprising. I deeply regret this situation.
"I thank all those who have supported me and I am most grateful. The operation of the club will continue with other stakeholders to prepare for the new season."