You need to place the \foreach
loop inside the frame and use overlay specifications for the \draw
command (this will produce several slides inside a single frame, which then will give you the animation effect); something along these lines:
\documentclass{beamer}
\usepackage{tikz}
\tikzset{
invisible/.style={opacity=0,text opacity=0},
visible on/.style={alt=#1{}{invisible}},
alt/.code args={<#1>#2#3}{%
\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}}
},
beameralert/.style={alt=<#1>{fill=red!30,rounded corners,inner sep=1pt}{},anchor=base},
}
\begin{document}
\begin{frame}
\centering
\begin{tikzpicture}[scale=3]
\foreach \x in {0,...,10}
{
\draw[
color=orange,
domain=0:\x,
samples=200,
visible on=<\number\numexpr\x+1\relax->
]
plot (canvas polar cs:angle=\x r,radius= {\x});
}
\end{tikzpicture}
\end{frame}
\end{document}

I used the visible on
style (illustrated, for example, in this answer) for the overlay specification to prevent the elements from jumping around.
In the above solution, the whole spiral (the "old part" and the new one) is drawn in each slide, so the compilation time can be very long and the calculations will be increasingly heavy; to prevent this you can use
\documentclass{beamer}
\usepackage{tikz}
\tikzset{
invisible/.style={opacity=0,text opacity=0},
visible on/.style={alt=#1{}{invisible}},
alt/.code args={<#1>#2#3}{%
\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}}
},
beameralert/.style={alt=<#1>{fill=red!30,rounded corners,inner sep=1pt}{},anchor=base},
}
\begin{document}
\begin{frame}
\centering
\begin{tikzpicture}[scale=3]
\xdef\lastx{0}
\foreach \x in {1,...,20}
{
\draw[
color=orange,
domain=\lastx:\x,
samples=200,
visible on=<\number\numexpr\x+1\relax->
]
plot (canvas polar cs:angle=\x r,radius= {\x});
\xdef\lastx{\x}
}
\end{tikzpicture}
\end{frame}
\end{document}
so in each new slide only the new piece is added. The result:

And here's a version of the animation produced using the animate
package; only some pdf viewers (Acrobat Reader, tipically) will support the animation correctly:
\documentclass{beamer}
\usepackage{tikz}
\usepackage{animate}
\usepackage{ifthen}
\hypersetup{pdfpagemode=FullScreen}
\newcounter{tmp}
\stepcounter{tmp}
\begin{document}
\begin{frame}[fragile]{}
\begin{center}
\begin{animateinline}[loop,poster=first,controls]{2}
\whiledo{\thetmp<21}{%
\begin{tikzpicture}
\begin{scope}[scale=2]
\foreach \x in {1,...,\thetmp}
{
\draw[
color=red,
domain=0:\thetmp,
samples=200,
]
plot (canvas polar cs:angle=\x r,radius= {\x});
}
\end{scope}
\path[use as bounding box] (-3,-3) rectangle (3,3);
\end{tikzpicture}%
\stepcounter{tmp}
\ifthenelse{\thetmp<21}
{\newframe}
{\end{animateinline}\relax}
}
\end{center}
\end{frame}
\end{document}