Is it permissible for a female student to examine a male patient for the necessity of an exam, if that requires touching him, and otherwise she would not be able to take the exam?

It is not permissible for a woman to touch the body of a non-mahram man, whether for teaching or examination, unless there is an extreme necessity, and if it is not possible for a substitute woman or one of his mahrams to perform the examination. The basic principle is the obligation to cover the awrah, but it is permissible to uncover it and look at it if there is a need or a legitimate interest, such as training medical students to provide Muslim doctors. If a woman is compelled, she must fear Allah and wear gloves to prevent direct touching. It is permissible for a woman to treat a non-mahram man when necessary, but without direct touching if possible.