What is the ruling on placing a thermometer moistened with disinfectant solution and a tongue depressor in the mouth of a fasting person for a medical examination, and inserting a finger moistened with water into the mouth out of necessity to remove a foreign object or food residue? And is it necessary to dry it before insertion?

Fasting is not invalidated by things reaching the mouth unless something from them reaches the internal cavity (jawf). Accordingly, the aforementioned things do not invalidate the fast as long as nothing from them reaches the internal cavity. What is obligatory is to be cautious about the wetness of water reaching the throat and then the internal cavity, because a fasting person is allowed to rinse their mouth but should not exaggerate, lest any water precede them to the internal cavity.