Yes, "straight" lines on a manifold are geodesics, which obey the geodesic equation $$\frac{\text{d}^2x^\rho}{\text{d}\tau^2} = -\Gamma^\rho_{\mu\nu}\frac{\text{d}x^\mu}{\text{d}\tau}\frac{\text{d}x^\nu}{\text{d}\tau}$$
On the surface of a sphere, geodesics are great circles. If the frictionless track is a great circle, the object will travel along it without feeling any force (except the normal force). Otherwise, the track will apply a force in order to keep the object moving along it.
Note that the geodesics on the surface of the Earth (great circles) are not the same as geodesics in spacetime, which correspond to test particles in free fall.