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
defprogressbar@progressbar{} % the progress bar
newcountprogressbar@tmpcounta% auxiliary counter
newcountprogressbar@tmpcountb% auxiliary counter
newdimenprogressbar@pbht %progressbar height
newdimenprogressbar@pbwd %progressbar width
newdimenprogressbar@tmpdim % auxiliary dimension

progressbar@pbwd=paperwidth
progressbar@pbht=1cm

% the progress bar
defprogressbar@progressbar{%
    progressbar@tmpcounta=insertframenumber
    progressbar@tmpcountb=inserttotalframenumber
    progressbar@tmpdim=progressbar@pbwd
    % to show current values
    rlap{theprogressbar@tmpdim, theprogressbar@tmpcounta}%
    pgfmathparse{progressbar@tmpcounta/progressbar@tmpcountb}%
    progressbar@tmpdim=pgfmathresultprogressbar@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
newdimenprogressbar@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}