How do I change the text color for a part of an equation after a pause in beamer?

You can use the \only overlay specification:

\documentclass[mathserif]{beamer}
\usetheme{Copenhagen}
\usepackage{color,xcolor,ucs}
\usepackage{beamerthemesplit}
\usepackage{graphicx}
\usepackage{amsmath,amsfonts,amssymb,setspace,mathtools}
\begin{document}
\section{Introduction}
\frame{\frametitle{NPDE}
\begin{itemize}
    \pause
  \item item 1 here
    \pause
  \item item 2 here
    \pause
  \begin{equation*}
          \frac{\partial u}{\partial t}= \only<4>{\textcolor{red}{u(n,t)}u(n+1,t)}\only<5>{u(n,t)\textcolor{red}{u(n+1,t)}} -u(n,t)u(n-1,t)
  \end{equation*}
\end{itemize}
}
\end{document}

And here are the fourth and fifth slides showing the relevant equation:

I know this is an old question, but the following is a bit more straightforward than the other answer provide (IMO).

Just add an overlay specification to the \textcolor command:

\documentclass[mathserif]{beamer}
\usetheme{Copenhagen}
\usepackage{color,xcolor,ucs}
\usepackage{beamerthemesplit}
\usepackage{graphicx}
\usepackage{amsmath,amsfonts,amssymb,setspace,mathtools}
\begin{document}
\section{Introduction}
\frame{\frametitle{NPDE}
\begin{itemize}
  \item<2-> item 1 here
  \item<3-> item 2 here
  \begin{equation*}
          \frac{\partial u}{\partial t}= \textcolor<3>{red}{u(n,t)}\textcolor<4>{red}{u(n+1,t)} -u(n,t)u(n-1,t)
  \end{equation*}
\end{itemize}
}
\end{document}