Not working for me either, and I've applied and checked the rotation on both the circle and the cone object. Project mode is doing nothing at all, just leaves the circle unmoved in space. (I'm using blender 2.79b.)
Edit: Found the problem - needed to tick the X-axis box on the shrinkwrap modifier controls.
Reapplying the rotation helps me, but I dont understand the undelying principle. Would you mind explaining?