Beamer: use glyphs from smaller font size, but enlarge

Consider this minimal example that uses the default of 11pt font and a 43 (or 4:3 aspect ratio, or 12.8cm x 9.6cm page dimension):

documentclass{beamer}
  
usetheme{Warsaw}

begin{document}

begin{frame}
  frametitle{There Is No Largest Prime Number - 11pt}
  framesubtitle{The proof uses textit{reductio ad absurdum}.}
  begin{theorem}
    There is no largest prime number.
  end{theorem}
  begin{proof}
    begin{enumerate}
      item<1-| alert@1> Suppose $p$ were the largest prime number.
      item<2-> Let $q$ be the product of the first $p$ numbers.
      item<3-> Then $q+1$ is not divisible by any of them.
      item<1-> But $q + 1$ is greater than $1$, thus divisible by some prime number not in the first $p$ numbers.qedhere
    end{enumerate}
  end{proof}
end{frame}

end{document}

You can choose the 8pt font option, and resize the page to have a similar aspect ratio, but slightly smaller (by a scaling factor of 8/11 thanks to xfp):

documentclass[8pt]{beamer}

usepackage{xfp}
makeatletter
% Taken from beamer.cls' default geometry settings
% http://mirrors.ctan.org/macros/latex/contrib/beamer/base/beamer.cls
geometry{%
  papersize={fpeval{beamer@paperwidth*8/11}pt,fpeval{beamer@paperheight*8/11}pt},
  hmargin=fpeval{8/11}cm,% 1cm
  vmargin=0cm,%
  head=fpeval{0.5*8/11}cm,% 0.5cm
  headsep=0pt,%
  foot=fpeval{0.5*8/11}cm% 0.5cm
}
makeatother
  
usetheme{Warsaw}

begin{document}

begin{frame}
  frametitle{There Is No Largest Prime Number - 8pt}
  framesubtitle{The proof uses textit{reductio ad absurdum}.}
  begin{theorem}
    There is no largest prime number.
  end{theorem}
  begin{proof}
    begin{enumerate}
      item<1-| alert@1> Suppose $p$ were the largest prime number.
      item<2-> Let $q$ be the product of the first $p$ numbers.
      item<3-> Then $q+1$ is not divisible by any of them.
      item<1-> But $q + 1$ is greater than $1$, thus divisible by some prime number not in the first $p$ numbers.qedhere
    end{enumerate}
  end{proof}
end{frame}

end{document}

Here's a rough visual of the difference:

You may have to change other lengths as well just because the scaling doesn't translate unilaterally to font-related concepts. Visually, however, it suffices to be very similar in display.

The simplest way to do this is to use the OpticalSize= font feature, from fontspec, and a font that supports it, such as Latin Modern or the TeX Gyre series. This has none of the side-effects of shrinking the document size.

There is really no reason not to use modern software with beamer, since you aren’t submitting a beamer presentation to a publisher who requires you to use PDFTeX.

documentclass{beamer}
tracinglostchars=2
usetheme{Warsaw}
usefonttheme{professionalfonts}
usepackage{unicode-math}

defaultfontfeatures{ Ligatures=TeX, OpticalSize=20 }

setmainfont{Latin Modern Roman}
setsansfont{Latin Modern Sans}
setmathfont{Latin Modern Math}
setmathfont{XITS Math}[range=QED, Scale=MatchUppercase]

renewcommandqedsymbol{ensuremath{QED}}

begin{document}

begin{frame}
  frametitle{There Is No Largest Prime Number - 20pt}
  framesubtitle{The proof uses textit{reductio ad absurdum}.}
  begin{theorem}
    There is no largest prime number.
  end{theorem}
  begin{proof}
    begin{enumerate}
      item<1-| alert@1> Suppose $p$ were the largest prime number.
      item<2-> Let $q$ be the product of the first $p$ numbers.
      item<3-> Then $q+1$ is not divisible by any of them.
      item<1-> But $q + 1$ is greater than $1$, thus divisible by some prime number not in the first $p$ numbers.qedhere
    end{enumerate}
  end{proof}
end{frame}

end{document}

And mutatis mutandis:

Additional use of setmathfont[range=... can get you sans-serif math letters, too.