Why is there an arithmetic overflow in the progress bar with more than 44 frames?

TeX has a \maxdimen (16383.99999pt), which represents the largest dimension you can use in an dimension expression. See discussions among \maxdimen on this site, including my answer.

When \x == 45,

• \progressbar@tmpdim == 364.19536pt and \progressbar@tmpcounta == 45,
• their multiplication 364.19536pt * 45 = 16,388.7912pt, is slightly larger than \maxdimen, hence raises "arithmetic overflow" and the result of multiplication is truncated.

Since the final result @tmpdim * @tmpcounta / @tmpcountb is smaller than \maxdimen, we can firstly compute @tmpcounta / @tmpcountb, then compute @tmpdim * <ratio>. Here is a try making use of \pgfmathparse from pgfmath, which is an autoloaded sub-package of tikz:

\documentclass{beamer}

\usepackage{tikz}

\makeatletter
\def\progressbar@progressbar{} % the progress bar
\newcount\progressbar@tmpcounta% auxiliary counter
\newcount\progressbar@tmpcountb% auxiliary counter
\newdimen\progressbar@pbht %progressbar height
\newdimen\progressbar@pbwd %progressbar width
\newdimen\progressbar@tmpdim % auxiliary dimension

\progressbar@pbwd=\paperwidth
\progressbar@pbht=1cm

% the progress bar
\def\progressbar@progressbar{%
    \progressbar@tmpcounta=\insertframenumber
    \progressbar@tmpcountb=\inserttotalframenumber
    \progressbar@tmpdim=\progressbar@pbwd
    % to show current values
    \rlap{\the\progressbar@tmpdim, \the\progressbar@tmpcounta}%
    \pgfmathparse{\progressbar@tmpcounta/\progressbar@tmpcountb}%
    \progressbar@tmpdim=\pgfmathresult\progressbar@tmpdim
    \begin{tikzpicture}
    \useasboundingbox (0pt, 0pt) rectangle ++ (\progressbar@pbwd, \progressbar@pbht);
    \begin{scope}
      \clip (\progressbar@tmpdim, 0pt) rectangle (\progressbar@pbwd, \progressbar@pbht);
      \node[anchor=south west,inner sep=0pt,outer sep=0pt,minimum height=1cm,minimum width=\paperwidth,fill=green] at (0pt,0pt) {};
    \end{scope}
    \end{tikzpicture}%
}

\addtobeamertemplate{footline}{}{\vspace*{-1cm}\progressbar@progressbar}
\makeatother

\begin{document}
\foreach \x in {1,2,...,45} {\begin{frame}[label=test]{My frame}
    Test \x
\end{frame}}

\end{document}

Update

Sorry, I paid too much attention to the error message "arithmetic overflow" yesterday, but not the whole picture of drawing a progress bar. Inspired by @TobiBS's answer, I find the implementation can be further simplified to

\documentclass{beamer}
\usepackage{tikz}

\makeatletter
\newdimen\progressbar@height
\progressbar@height=1cm

\addtobeamertemplate{footline}{}{%
  \begin{tikzpicture}
    \useasboundingbox (0pt, 0pt) rectangle (\paperwidth, \progressbar@height);
    % you can use pgfmath expressions directly in coordinate specifications
    \fill[green] (\insertframenumber/\inserttotalframenumber*\paperwidth, 0)
       rectangle (\paperwidth, \progressbar@height);
  \end{tikzpicture}%
}
\makeatother

\begin{document}
\foreach \x in {1,2,...,50} {
  \begin{frame}{My frame}
    Test \x
  \end{frame}
}
\end{document}