What happens to minerals once you boil off liquid in a water and fat mixture?
Because dietary minerals are predominantly water soluble and not significantly soluble in fats, your first statement is correct; the minerals will not dissolve in the fats as you boil away the water.  This is true even after all the water has boiled off, so the answer to your second question is no. As the water boils away, the minerals will precipitate out until no water remains. At that point, you will predominantly have the minerals at the bottom of the pan, covered in the fats. It is unlikely any of the minerals will have a low enough density to float on top of the fats. With mixing, you may be able to keep a suspension of the minerals within the fats even though they don't technically dissolve.  

It will probably be difficult to tell exactly when all of the water is gone. There should be a difference in the appearance of the way the stuff boils once the water is gone and it's just fats boiling, but this could be a subtle change. You might be able to tell when the water is gone by monitoring the temperature of the mixture. It should stay close to the boiling point of water until the water is gone and then increase when you just have the fats boiling.  

I'm far from a chef, so I hope I haven't made any improper assumptions on that front. Don't hesitate to ask for clarifications in the comments.