How to Get Vertical Timeline work in Beamer?

The "problem" is, that beamer already loads xcolor and your [x11names] causes an option clash. Fortunately beamer has an own build-in option.

\documentclass[xcolor=x11names,table]{beamer}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage[TS1,T1]{fontenc}
\usepackage{fourier, heuristica}
\usepackage{array, booktabs}
%\usepackage{graphicx}
%\usepackage[x11names]{xcolor}
%\usepackage{colortbl}
\usepackage{caption}
\DeclareCaptionFont{blue}{\color{LightSteelBlue3}}

\newcommand{\foo}{\color{LightSteelBlue3}\makebox[0pt]{\textbullet}\hskip-0.5pt\vrule width 1pt\hspace{\labelsep}}

\begin{document}

    \begin{frame}

\begin{table}
\renewcommand\arraystretch{1.4}\arrayrulecolor{LightSteelBlue3}
\captionsetup{singlelinecheck=false, font=blue, labelfont=sc, labelsep=quad}
\caption{Timeline}\vskip -1.5ex
\begin{tabular}{@{\,}r <{\hskip 2pt} !{\foo} >{\raggedright\arraybackslash}p{5cm}}
\toprule
\addlinespace[1.5ex]
1933 & LCMV, aseptic meningitis. Epidemic St. Loius encephalitis. \\
1956 & Tacaribe virus.  \\
\end{tabular}
\end{table}
\end{frame}
\end{document} 

This answer uses \PassOptionsToPackage{x11names}{xcolor} following @Herbert's answer (LaTeX Error: Option clash for package xcolor). The caption is in a minipage environment to align it with the vertical timeline.

\PassOptionsToPackage{x11names}{xcolor}
\documentclass{beamer}

\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage[TS1,T1]{fontenc}
\usepackage{fourier, heuristica}
\usepackage{array, booktabs}
\usepackage{graphicx}
%\usepackage[x11names]{xcolor}
\usepackage{colortbl}
\usepackage{caption}
\DeclareCaptionFont{blue}{\color{LightSteelBlue3}}

\newcommand{\foo}{\color{LightSteelBlue3}\makebox[0pt]{\textbullet}\hskip-0.5pt\vrule width 1pt\hspace{\labelsep}}

\begin{document}

    \begin{frame}

\begin{table}
\renewcommand\arraystretch{1.4}\arrayrulecolor{LightSteelBlue3}
\captionsetup{singlelinecheck=false, font=blue, labelfont=sc, labelsep=quad}

\begin{minipage}{7cm}
    \caption{Timeline}
\end{minipage}
\vskip -1.5ex

\begin{tabular}{@{\,}r <{\hskip 2pt} !{\foo} >{\raggedright\arraybackslash}p{5cm}}
\toprule
\addlinespace[1.5ex]
1933 & LCMV, aseptic meningitis. Epidemic St. Loius encephalitis. \\
1956 & Tacaribe virus.  \\
\end{tabular}
\end{table}
\end{frame}
\end{document}