How to make \left, \right pairs of delimiter work over multiple lines?

You can't.

You can use some thing like \biggl, \biggr, or \left.\vphantom{...}\right). For example:

\[
\begin{split}
a &= \left( \frac12 + \frac13 + \frac14 \right. \\
  &\quad \left. {}+ a + b + c \vphantom{\frac12}\right)
\end{split}
\]

The solution for me was to use virtual dot delimiters

\frac{\partial F}{\partial x} &= \left[ \frac{\partial y}{\partial z} \right. \\
                              &* \left. \frac{\partial z}{\partial x} \right]

which was the comment posted by percusse in this question which was marked as a duplicate.

The virtual dot delimiters are in Leo Liu's answer above, but a TeX n00b like me wouldn't have seen it.

FYI Sphinx-1.1.3 already supports multi-line math with the math directive using the AmSMath LaTeX package, i.e. breqn is not necessary.

The breqn package allows for automatic line breaking of equations, and for having \left and \right on different lines.