plot() pretty much creates lines with Z coordinate 0. surf() is going to be using 3D coordinates. Unless your surface happens to be below z 0, your plot lines will be below (at least part of) the surface.
If you want to draw lines right on the surface, then you would need to use z coordinates appropriate for each location when you constructed the line. If your line is a coarser resolution than your surface, this might require interpolating the line at each grid location in order to get the z right for each place.
What you might possibly find easier is to use texture mapping of an image containing the lines, onto the surface. This approach won't be much fun either, though.
Anyhow, remember: if there are two objects both in 3D space, then the object that is closer to the eye is the one that is going to be visible. (lines are implicitly 3D space even when using plot() instead of plot3())
If two three-space are equally close to the eye, then the precedence depends on which renderer you are using. zbuffer and painters will render in order, newest object on top. opengl, however, will ignore the order, and is defined to render lines and surfaces in a particular order (and you'll probably want the exact opposite order at some point... which might be what your graphics driver gives to you as some exactly reverse the defined order!)
