No, your solutions do not need to be closed curves. For example, take the vector field "multiplication by $i$" on the $3$-sphere, thought of as the sphere of unit quaternions. That has closed solutions (Hopf fibration fibres). With a little bump function you can perturb this vector field to ensure there's only two closed solutions -- make the perturbation "away" from one closed solution and "towards" the other. This can be made concrete in many ways but I hope you get the idea.
edit: If you're interested in minimizing the number of closed orbits you can readily get down to one, on $S^3$. The meta-idea above is that the normal bundle to the Hopf fibration is the pull-back of the tangent bundle to $S^2$. $S^2$ has a flow with only one fixed point, so pull that vector field back to the normal bundle of the Hopf fibration, add it to "multiplication by $i$" and now you have a vector field with only one closed orbit (the Hopf fibre over the fixed point to your vector field on $S^2$).
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