Is Pauling electronegativity well-defined?
If you have a single, unique value of $\chi_i$ for each element $i$ then yes, the sum of the three values is always zero. 

If you are assigning electronegativities that differ with oxidation state or chemical environment, the answer can be no (for example, if element B has a different electronegativity in an AB bond than it has in a BC bond).

The Pauling electronegativity difference is

$${\chi}_{a}-{\chi}_{b}=(1\ eV)^{-\frac{1}{2}}\,\sqrt{{E}_{d}\left( a,b\right) -\frac{{E}_{d}\left( a,a\right) +{E}_{d}\left( b,b\right) }{2}}$$

where $E_d(i,j)$ is the dissociation energy for a single bond between element $i$ and element $j$.
The definition assumes that you can consistently and accurately say that the square of the electronegativity difference between A and B is equal to the difference between the dissociation energy for AB and the average of the dissociation energies for AA and BB. If this isn't consistently true for each combination of three elements A, B, and C, your sum will be nonzero. Pauling fit his electronegativity values to the data to average out these inconsistencies.