ParametricPlot3D[ {(2 + Cos[ 2 Pi v]) Cos[
2 Pi u], (2 + Cos[ 2 Pi v]) Sin[2 Pi u], Sin[2 Pi v]}, {v, 0, 1}, {u, 0, 1}]
You want to draw certain lines on above torus it appears.
Next,please describe the character of winding lines you choose to depict on it, as that comes at first. For example do you want to draw geodesics? loxodromes? Planar cut loci? Intersection with another surface? or what? Animation can come later on.
$u(t), v(t)$ is not specific enough. The first real part of exponential is not clear. Can you put that the line parametrization into the form:
$$ x(t),y(t),z(t) ? $$
EDIT1:
Geodesics can be found by employing Clairaut's Law
$$ r\cdot \sin \alpha = r_{min}$$
Useful reference
TorusGeodesicsRef
If inner radius is $c$ then there are three possible scenarios for return of running geodesics.
If $ r_{min} < c $ geodesics cover the torus fully, criss crossing but running in same direction.
If $ r_{min} > c $ geodesics cover outside portion only, criss crossing while running to and fro.
If $ r_{min} = c $ geodesics asymptotically reach $ r_{min} = c $ towards an inner torus equator line of no return ... at the torus inner equatorial radius.
EDIT2:
Since you left the choice of type of line to be drawn I chose edge lines of constant width strips. They can be used to wind closely packed copper wires for example on a toroidal solenoid. They are also geodesic parallel lines because separation between them is constant satisfying DE $ r\cdot \cos \alpha = const.$ They are orthogonal trajectories to geodesics.

So Viviani can join Sumit's bandwagon ! ( as $ u=v ?$ )
sph = {Sin[v] Cos[u], -Cos[v] Cos[u], Sin[u]};
cyl = {.5 Sin[p], -.5 - .5 Cos[p] , q}
pl0 = ParametricPlot3D[cyl, {p, 0, 2 Pi}, {q, -1, 1}, Mesh -> False, PlotStyle -> Opacity[0.3]];
pl1 = ParametricPlot3D[sph, {v, 0, 2 Pi}, {u, 0, 2 Pi}, Mesh -> False,
PlotStyle -> Opacity[0.7]];
(* u=v=t for Viviani_Curve *)
pl2 = ParametricPlot3D[Block[{u = t, v = t}, sph], {t, 0, 2 Pi}, PlotStyle -> Thick];
Animate[Show[pl1, pl0, pl2, Graphics3D[{Red, Sphere[Block[{u = tz, v = tz}, sph], 0.04]}]], {tz, 0, 2 Pi}]