Is it obligatory to make up the fast of a day on which one went to a male dentist's clinic, and he forcefully opened the woman's mouth with his hand during the examination?

What your friend did does not invalidate her fast, nor does it necessitate a make-up fast for her. However, it is not permissible for a woman to seek treatment from a man if it requires him to see any part of her 'awrah, unless she cannot find a woman to treat her, or there is a compelling necessity for that treatment.