What is the ruling of the religion on a dentist touching a female patient's hand or opening her mouth during treatment without an immoral intention?

It is permissible for a male doctor to look at or touch what is necessitated by the need to treat and examine a foreign (non-mahram) woman. Similarly, it is permissible for a female doctor to do so with a foreign (non-mahram) man. The details and conditions for this are available in Fatwa no. 33861.