A series of claims were made against the pair in a New York Times article.
Weber denies the claims, while lawyers for Testino said his accusers "cannot be considered reliable sources".
Vogue publisher Conde Nast said it would not work with the men "for the foreseeable future".
Artistic director Anna Wintour, who is also Vogue's editor-in-chief, said in a statement that the claims had "been hard to hear and heartbreaking to confront".
