Triple integral in mathpazo

The mathpazo package, which provides Palatino-clone text and math fonts, hasn't been updated in a long time. As you've discovered, it also lacks macros to generate double- and triple-integral symbols.

An up-to-date alternative to mathpazo is the newpxtext and newpxmath pair of packages. A side-benefit of loading newpxmath is that it provides a \iiint macro by default -- as well as several other, compatibly styled integral symbols.

Since newpxmath loads the amsmath package automatically (if it hasn't been loaded already), there's no need to load amsmath in separate instruction. Likewise, unless the noamssymbols option is set, the newpxmath package loads the pxfonts version of amssymb -- no need to load the amssymb package separately either

\documentclass{article}
\usepackage{newpxtext,newpxmath}
\begin{document}
$\int \oint \iint \iiint \displaystyle\int \oint \iint \iiint$

The quick brown fox jumps over the lazy dog.
\end{document}

---

Addendum, posted after the OP provided a full MWE: The newpxmath and newpxtext packages must be loaded after amsmath.

\documentclass[a4paper]{memoir}
\usepackage[T1]{fontenc}
%\usepackage{fouriernc} % not compatible with Palatino
\usepackage{amsmath}    % is loaded automatically by 'newpxmath' package
%\usepackage{amssymb}   % no need to load this package either
\usepackage{amsthm}                 
\usepackage{mathrsfs}   
\usepackage{newpxtext,newpxmath}

\newcommand*\dif{\mathop{}\!\mathrm{d}}                         

\begin{document}
\setcounter{chapter}{2} % just for this example
\section{A Section}

This should be a triple integral:
\begin{equation} \label{eqn:triple_integral}
\iiint\limits_V \! \dif V \, (\nabla\phi)^2 = 0.
\end{equation}
\end{document}

Using xelatex or lualatex makes things easier:

\documentclass[a4paper]{memoir}
\usepackage{amsthm}                 
\usepackage{unicode-math}
\setmainfont{TeX Gyre Pagella}
\setmathfont{TeX Gyre Pagella Math}

\newcommand*\dif{\mathop{}\!\mathrm{d}}                         

\begin{document}
    \setcounter{chapter}{2} % just for this example
    \section{A Section}
    This should be a triple integral:
    \begin{equation} \label{eqn:triple_integral}
    \iiint\limits_V \! \dif V \, (\nabla\phi)^2 = 0.
    \end{equation}
\end{document}

The error is in loading fouriernc, which is for using New Century Schoolbook, rather than Palatino. Load either it or mathpazo, not both.

The centertags option is active by default for amsmath.

\documentclass[a4paper,10pt,reqno,oldfontcommands]{memoir}
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}

\newcommand*\dif{\mathop{}\!\mathrm{d}}

\begin{document}

\section{A Section}

This should be a triple integral:
\begin{equation}
\iiint\limits_V (\nabla\phi)^2 \dif V = 0.
\label{eqn:triple_integral}
\end{equation}

\end{document}

A different option is to use newpx (amsthm should be loaded before it) and amssymb should be removed.

\documentclass[a4paper,10pt,reqno,oldfontcommands]{memoir}
\usepackage[T1]{fontenc}
\usepackage{amsthm}
\usepackage{newpxtext,newpxmath}
\usepackage{mathrsfs}

\newcommand*\dif{\mathop{}\!\mathrm{d}}

\begin{document}

\section{A Section}

This should be a triple integral:
\begin{equation}
\iiint\limits_V (\nabla\phi)^2 \dif V = 0.
\label{eqn:triple_integral}
\end{equation}

\end{document}