Summing over conditional probabilities

The equation does certainly not apply in general. What you can write however is \begin{equation} P(a|b) = \sum_z P(a,z|b), \end{equation} which is sometimes referred to as marginalization. Note that the above equation simply describes how to go from a joint probability mass function $P(x,y)$ to the probability mass function $P(x)$ (or $P(y)$), that is, by summing out the other variable. (Similar arguments apply to probability density functions by replacing the above summation with integration.)

Now, one can apply Bayes rule and further expand $P(a,z|b) = P(a|z,b) P(z|b)$ to get\begin{equation} P(a|b) = \sum_z P(a|z,b) P(z|b). \end{equation} The formula that you posted simply drops the conditioning on $b$ in the first factor (implying $P(a|z,b) = P(a|z)$), which cannot be done in general.