amsmath's multline causes "Missing $ inserted" error

The root cause of the problem you've encountered is that you're trying to use a text-mode-only command -- d -- in math mode.

Rather than undefine and then redefine d suitably, I think it's better to create a macro with a new name -- say, diff -- to denote the "differential operator". E.g.,

newcommand{diff}{mathop{}!mathrm{d}}

Observe that this command does not take an argument. Thus, I would encourage you to just write diff S, not diff{S}, as the latter might create the (misleading and inappropriate) impression that diff is a macro that takes an argument.

I would also replace with all instances of intlimits with just int, as I can't see a (typographic) justification for elongating the equation in the vertical direction.

Finally, I think the equation is easier to read if you used an equation/aligned combination instead of a multline environment.

documentclass{article} % or some other suitable document class
usepackage{amsmath}
newcommand{diff}{mathop{}!mathrm{d}} % don't re-purpose the existing 'd' macro
newcommand{gnpn}{Gammanablamkern-2mu phicdotmathbf{n}} % handy shortcut macro
begin{document}

begin{equation}label{eq:volume_boundaries}
begin{aligned}[b]
int_S gnpn diff S 
    &=      int_{S_mathrm{n}}!  (gnpn)_{mathrm{n}}diff S_{mathrm{n}} 
           +int_{S_mathrm{e}}!  (gnpn)_{mathrm{e}}diff S_{mathrm{e}}
    &quad +int_{S_mathrm{s}}!  (gnpn)_{mathrm{s}}diff S_{mathrm{s}}
           +int_{S_mathrm{w}}!!(gnpn)_{mathrm{w}}diff S_{mathrm{w}}
end{aligned}
end{equation}
end{document}