How to shade a polygon with curved lines in tikz?

I don't know how you wanted it shaded, but if you draw it as a single path you can use for example \shadedraw[left color=.., right color.., ].

\documentclass{article}
\usepackage{standalone}
\usepackage{tikz}
\begin{document}
    \begin{tikzpicture}
        \draw [->] (0, 0) -- (5, 0) node [right] {$X$};
        \draw [->] (0, 0) -- (0, 3) node [above] {$Y$};

        \draw [domain=0.66:4.5, thick, blue] plot (\x, {2/\x});
        \draw [domain=1.66:4.5, thick, blue] plot (\x, {6/\x});

        \shadedraw[
            draw=red,
            ultra thick,
            left color=blue,
            right color=red,
            bend angle=10
          ]
          plot[domain=2.5:3.5] (\x, {2/\x})
          to[bend left] (3,2)
          plot[domain=3:2] (\x, {6/\x})
          to[bend right] (2.5, 2/2.5);

        \node [above] at (2, 3) {$C$};
        \node [below] at (2.5, 2/2.5) {$B$};
        \node [above] at (3, 2) {$D$};
        \node [below] at (3.5, 2/3.5) {$A$};
    \end{tikzpicture}
\end{document}

Using fadings library of TiKz:

\documentclass[tikz,margin=5mm]{standalone}
\usetikzlibrary{fadings}
\makeatletter
\pgfdeclareverticalshading{pgf@lib@fade@north}{100bp}
{color(0bp)=(pgftransparent!0);
 color(5bp)=(pgftransparent!10);
 color(60bp)=(pgftransparent!100); color(80bp)=(pgftransparent!100)}%
\pgfdeclarefading{myfade}{%
  \pgfuseshading{pgf@lib@fade@east}%
}
\makeatother
\usepackage{tikz}
\begin{document}
    \begin{tikzpicture}
        \draw [->] (0, 0) -- (5, 0) node [right] {$X$};
        \draw [->] (0, 0) -- (0, 3) node [above] {$Y$};

        \draw [domain=0.66:4.5, thick, blue] plot (\x, {2/\x});
        \draw [domain=1.66:4.5, thick, blue] plot (\x, {6/\x});

        \draw [red, ultra thick] (3, 2)coordinate(D) edge [bend right=10] (3.5, 0.57)coordinate(A);
        \draw [red, ultra thick] (2, 3)coordinate(C) edge [bend right=10] (2.5, 0.8)coordinate(B);

        \draw [domain=2.5:3.5, ultra thick, red] plot (\x, {2/\x});
        \draw [domain=2:3, ultra thick, red] plot (\x, {6/\x});

        \node [above] at (2, 3) {$C$};
        \node [below] at (2.5, 0.8) {$B$};
        \node [above] at (3, 2) {$D$};
        \node [below] at (3.5, 0.57) {$A$};
        \fill[red,path fading=myfade]  (3,2) to [bend right=10] (3.5,0.57) 
to [bend left=10] (2.5,0.8) to [bend left=10] (2,3) to [bend right=10] (3,2);

    \end{tikzpicture}
\end{document}