There is a constitutional role in this for the military as well. It is clear in the oath taken that was already pointed out here...
I__________do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; and that I will obey the orders of the President of the United States and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice. So help me God." (Title 10, US Code; Act of 5 May 1960 replacing the wording first adopted in 1789, with amendment effective 5 October 1962)
Now, I am not in the military, but what I have gleaned from people I know in the military is that there is a priority to those various oaths. So, above "obeying the POTUS" is "supporting and defending the Constitution". (If I am wrong the priority thing, please correct me).
If that is true, and if Obama is found to not be a natural born citizen, then the military is obligated to remove him from office if he refuses to leave (ala recent events in Honduras). Even if that priority thing is not true, the military would still be obligated to remove Obama from office if he refused to leave because he would not be the POTUS at that point.
As to the "questioning orders" thing; I cannot give you or any solider an order. Only qualified individuals (commanding officers) can do that. It is less then clear if Obama is Constitutionally qualified to be recognized in that role as Commander-in-Chief So it is a legitimate issue to bring up.