Not exactly the result you wanted, but it can be a starting point. It combines framed and tikz packages. The idea can lead to further improvements.

See http://www.texample.net/tikz/examples/framed-tikz/ (btw, I'm the author)

UPDATE: Playing with the idea of a fractal decoration, and also with shaded paper and blurred drop shadows, I got the following results. Currently the implementation still does not use framed, so it is restricted to the case in which the framed paragraph does not break among pages.

\documentclass[a5paper]{article}
\usepackage{lipsum}   % To generate test text 
\usepackage{framed}
\usepackage{tikz}
\usepackage[margin=1cm]{geometry}% for screen preview
\usetikzlibrary{decorations.pathmorphing,calc,shadows.blur,shadings}
\pgfmathsetseed{1} % To have predictable results
% Define a background layer, in which the parchment shape is drawn
\pgfdeclarelayer{background}
\pgfsetlayers{background,main}

% This is the base for the fractal decoration. It takes a random point between the start and end, and
% raises it a random amount, thus transforming a segment into two, connected at that raised point
% This decoration can be applied again to each one of the resulting segments and so on, in a similar
% way of a Koch snowflake.
\pgfdeclaredecoration{irregular fractal line}{init}
{
  \state{init}[width=\pgfdecoratedinputsegmentremainingdistance]
  {
    \pgfpathlineto{\pgfpoint{random*\pgfdecoratedinputsegmentremainingdistance}{(random*\pgfdecorationsegmentamplitude-0.02)*\pgfdecoratedinputsegmentremainingdistance}}
    \pgfpathlineto{\pgfpoint{\pgfdecoratedinputsegmentremainingdistance}{0pt}}
  }
}

% define some styles
\tikzset{
   paper/.style={draw=black!10, blur shadow, shade=bilinear interpolation,
                 lower left=black!20, upper left=black!15, upper right=white, lower right=black!10},
   irregular border/.style={decoration={irregular fractal line, amplitude=0.2},
           decorate,
     },
   ragged border/.style={ decoration={random steps, segment length=7mm, amplitude=2mm},
           decorate,
   }
}

% Macro to draw the shape behind the text, when it fits completly in the
% page
\def\tornpaper#1{
\tikz{
  \node[inner sep=1em] (A) {#1};  % Draw the text of the node
  \begin{pgfonlayer}{background}  % Draw the shape behind
  \fill[paper] % recursively decorate the bottom border
        decorate[irregular border]{decorate{decorate{decorate{decorate[ragged border]{
        ($(A.south east) - (0, random*5mm)$) -- ($(A.south west) - (0, random*5mm)$)
        }}}}}
        -- (A.north west) -- (A.north east) -- cycle;
  \end{pgfonlayer}}
}


\begin{document}
\noindent
\tornpaper{
\parbox{.9\textwidth}{\lipsum[11]}
}

\bigskip
\noindent
\tornpaper{
\parbox{.9\textwidth}{\lipsum[15]}
}
\end{document}

NOTE Some pdf viewers (eg. Sumatra) have issues to display the interpolated shading used in the last examples.

UPDATE See also the related question Torn paper: matching up the torn edges ^[1]

EDIT: Added color on request.

Here is an approach which builds on Marc van Dongen's answer to Simulating hand-drawn lines ^[1].

\documentclass{memoir}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\definecolor{paper}{RGB}{239,227,157}
\usepackage{lipsum}
\begin{document}
\begin{tikzpicture}[pencildraw/.style={ %
    decorate,
    decoration={random steps,segment length=2pt,amplitude=1pt}
    } %
]
\node[
preaction={fill=black,opacity=.5,transform canvas={xshift=1mm,yshift=-1mm}},
pencildraw,draw,fill=paper,text width=.8\textwidth,inner sep=5mm] 
{\lipsum[1]};
\end{tikzpicture} 
\end{document}

Output:

And yes, this does not really look like paper.

I think this would be a natural application of the tcolorbox ^[1] package.

In the answer below, I've used part of Jesper's answer ^[2] to decorate the tcolorbox environment; this approach permits an environment-based approach, so that you can use, for example,

\begin{tornpage}
    \lipsum
\end{tornpage}

Furthermore, the tcolorbox allows page breaks.

Here's a complete MWE which can be built upon.

% arara: pdflatex
% !arara: indent: {overwrite: on}
\documentclass{article}
\usepackage[many]{tcolorbox}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{shadows}
\definecolor{paper}{RGB}{239,227,157}
\usepackage{lipsum}
\newtcolorbox{tornpage}{%
    enhanced jigsaw, breakable, % allow page breaks
    frame hidden, % hide the default frame
    overlay={%
        \draw [
            fill=paper, % fill paper
            draw=paper!50!black, % boundary colour
            decorate, % decoration
            decoration={random steps,segment length=2pt,amplitude=1pt},
            drop shadow, % shadow
        ]
        % top line
        (frame.north west)--(frame.north east)--
        % right line
        (frame.north east)--(frame.south east)--
        % bottom line
        (frame.south east)--(frame.south west)--
        % left line
        (frame.south west)--(frame.north west);
    },
    % paragraph skips obeyed within tcolorbox
    parbox=false,
}
\begin{document}
\begin{tornpage}
    \lipsum[1]
\end{tornpage}
\begin{tornpage}
    \lipsum
\end{tornpage}
\end{document}