Is thermal conductivity additive? Suppose I have water in a beaker. I measured, its thermal conductivity by some means to be $\kappa_0$. Next, I added some salt to it so that it dissociate from it. Say,
$$AB\rightarrow A^++B^-$$
Now, I again measured the thermal conductivity and found $\kappa$. Can I say that the thermal conductivity contribution from the ions is $\kappa_0-\kappa$?

This is in the following context.
 A: In general, no.
First, it would not be expected to be the sum, but rather some kind of average. Otherwise, you could add fifty different materials to your mix and get a huge conductivity. So a weighted average of some kind might apply to some mixtures.
The factors affecting thermal condutivity will change in a mixture such that the result of the combination is not a simple average.
In your example, salt is presumably in a solid form before, where both the pure water and the mixture are liquids. Solids and liquids have very different behavior. Salty water has very different electrical conductivity to pure water. The density of the result will be different to the density of the components.
So no, the resultant thermal conductivity is not a simple average average of the components.
Even a mechanical mixture is troublesome. Consider two solids that you ground up into a fine powder, then mixed thoroughly, then compressed. The resultant thermal conductivity might be expected to be the average of the two, possibly weighted in some way to account for the relative amounts of each. But then you would need to account for the contact between grains in the mix. That is a very complicated business indeed, depending on the compression pressure, the nature of grinding, the temperature, and the deformation character of the solids. Just to name a few.
