Nice question -- it's something I've never thought about. I would conjecture that the axis of a right circular cone passes through the center of the conic (and therefore through its coincident foci) precisely when the conic is a circle (in a plane perpendicular to the axis). All I know for certain about the plane sections of a cone can be found in the Wikipedia article on Dandelin spheres, although I find the picture there a bit confusing. To see how the axis of the cone intersects a plane section, imagine rotating a plane about a line that intersects the axis at a right angle, starting with a plane perpendicular to the axis -- the ellipse of intersection becomes more and more elongated until in the limit it becomes a pair of intersecting lines. It seems to me that one focus moves slowly away from the axis while the other moves quickly.

Chris