According to this Ruth Marcus op-ed in the Washington Post, Rep. Jason Chaffetz verbally floated the idea of a mental fitness test for the President of the United States:
"What is to be done? In a meeting last week with The Post editorial board, Jason Chaffetz (R-Utah), chair of the House Oversight Committee, said he was weighing legislation to require presidents to undergo an independent medical examination, including for mental health. Chaffetz cautioned that he wasn’t “talking about some of the rhetoric that’s flying around” about Trump. Still, he said, “If you’re going to have your hands on the nuclear codes, you should probably know what kind of mental state you’re in.” "
I find that quite interesting. Because if Trump really pushes the wrong policy buttons (according to what Republicans want), there are several ways that they could come after him: threaten to impeach him, threaten to find him incompetent or incapacitated to be in office (the 25th Amendment maneuver), or the Chaffetz legislation. Presumably the legislation would have some kind of provision for temporary/permanent replacement in office if the POTUS is found to be mentally ill.