What are dipole moments in a molecule supposed to act upon? In a $\ce{CO_2}$ molecule, a total of four electron pairs are shared between the carbon and oxygen atoms, such that 2 pairs are shared between the carbon atom and each oxygen atom. Oxygen has a greater electronegativity hence the atoms of oxygen will spend greater time with shared pairs and are partially negative. Since both oxygen atoms will exhibit this same behavior, the carbon atom will become partially positive.

I was reading about the dipole moment of $\ce{CO2}$ (e.g., as here), in which it said that the net polarity of the $\ce{CO_2}$ molecule is zero because if we add the individual dipole moment vectors directed from the positive pole (carbon) to the negative poles (the oxygen atoms), the resultant will be zero which implies the polarity of the molecule is zero.

My problem is this:

If these individual dipole moments (vectors) are showing the opposite and equal pull of shared pairs of electrons toward more electronegative atoms (oxygen atoms), then the carbon atom should stay partially positive with the oxygens remaining partially negative.

Given this, though, it seems that the molecule should remain polar on whole.  This is because these dipole moment vectors are acting on different shared pairs: the first vector acts on the two shared electron pairs between the carbon atom and one of the oxygen atoms, and the other acts on the two shared electron pairs between the carbon atom and the other oxygen atom. According to the following law, then, these vectors cannot cancel each other and thus $\vec u \neq\vec 0$:


  Two equal and opposite vectors do not cancel each other out when acting on different bodies.


My best attempt to resolve this contradiction:

Since in reality it is the case that the net polarity of $\ce{CO_2}$ is null, I think that both vectors (viz., the 'pull' from both oxygen atoms) must be acting individually on all electrons of carbon atom and therefore they cancel each other out resulting in molecule to be non-polar on whole. This explanation isn't all that satisfying to me, however.
Your confusion appears to arise from trying to consider dipole vectors as representing forces, which is incorrect.

Dipole vectors don't "act on" anything; they describe the displacement of the "center-of-charge"&ddagger; of a given region of space (including both the electron cloud and the nuclei) from a particular reference point. In particular, I believe you are confusing dipole moment vectors with electric field vectors: 


The electric field vector at a point describes the force experienced there by a point charge
The dipole moment vector at a point describes the displacement of the "center of charge" of the system from that point


Thus, whereas the electric field contributes to the forces present in a system, the dipole is simply a post facto descriptor of a system.



Further, in this case the law you quoted is not applicable:


  Two equal and opposite vectors do not cancel each other out when acting on different bodies.


It only makes sense to calculate an overall dipole moment if you use the same reference point for all of the 'subsystem' dipole vectors. Thus, in this example of $\ce{CO2}$, the two $\ce{C-O}$ 'subsystem' vectors point in exact opposite directions from the same reference point, since $\ce{CO2}$ is linear.  Thus, since the outside atoms are both oxygens, the magnitudes of both 'subsystem' vectors are equal and their sum is the zero vector.



&ddagger;
For a non-electroneutral region of space, by center of charge I mean the charge-weighted centroid of the system: (1) Choose a reference point $\vec r_0$. (2) At each point $\vec r$ in the region of interest, multiply the charge at that point, $\rho\!\left(\vec r\right)$, by the distance from the reference point, $\vec r - \vec r_0$. (3) Integrate the resulting product, $\rho\!\left(\vec r\right)\cdot\left(\vec r - \vec r_0\right)$, over all space. (4) Divide by the net charge of the region of interest. Dipole moment calculations are more complicated for electroneutral systems, but the principle is similar.