Thanks for the reply, Frenchy Pilou. I redrew the object, but I'm still having the same problem.

When I turn the curves into planar surfaces, all render properly except for the two side curves (see the attached file.) Is this because the curve isn't all on the same plane? I also can't get it to extrude properly - it gives me a hollow extrusion. I'm guessing the side curves are my problem, but I have no clue as how to fix them.