 Solution

False. Vanishing torsion only controls each connected region on which the curvature is nonzero. A smooth curve can pass through a flat point or straight segment and emerge in a different plane.

For an explicit local construction on $(-1,1)$, put $\eta (s)=e^{-1/s^2}$ for $s\ne 0$ and $\eta (0)=0$, and define a unit tangent fieldThis field is smooth because $\eta$ is flat at zero. Its integral $\alpha (s)={\int }_0^{s}T(u)du$ is parametrized by arc length, has nonzero curvature for $s\ne 0$, and has zero torsion on both sides. Its negative half lies nontrivially in the $xy$-plane and its positive half lies nontrivially in the $xz$-plane, so the whole curve is not planar.