Dietary supplements maker Mannatech Inc. said it fired Grant Thornton LLP as its auditor after the accounting firm demanded that Mannatech remove its current chairman and founder from all responsibilities.

The move came two months after the chairman, Sam Caster, resigned as chief executive of the multilevel-marketing company amid pressure from shareholders and regulators that have accused the firm of using illegal sales tactics and boosting sales by making unfounded health claims for their products. Mannatech and Mr. Caster have strongly denied ...