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}
]

Have a look at the nath package.

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.

I had the same problem. I wanted to show this equation

begin{equation*}
    V_{sal} = -left(1.V_8 + frac{1}{2}.V_7 + frac{1}{8}.V_6 +frac{1}{8}.V_5  + frac{1}{16}.V_4 right.
     left. + frac{1}{32}.V_3 + frac{1}{64}.V_2 + frac{1}{128}.V_1right) 
end{equation*}

but since it is too long (in a two column document), I used multline enviroment but it gives an error (something like Missing right. inserted. end{multline*} or Extra right. end{multline*}). I solved it with this new code

begin{multline*}
V_{sal} = -left(1.V_8 + frac{1}{2}.V_7 + frac{1}{8}.V_6 +frac{1}{8}.V_5  + frac{1}{16}.V_4 right.
 left. + frac{1}{32}.V_3 + frac{1}{64}.V_2 + frac{1}{128}.V_1right) 
end{multline*}

Note that I added a right. at the end of the first row and left. at the beginning of the second row.