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I would like to be able to shade a torus in TikZ to make it look like a surface in 3D, similarly to how the sphere in this code is shaded to give it a 3D feel but the opacity is such that it does not look like a solid ball:

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

begin{tikzpicture}

draw (0,0) circle (1cm);

draw (-1,0) arc (180:360:1cm and 0.5cm);

draw[dashed] (1,0) arc (0:180:1cm and 0.5cm);

shade[ball color=blue!10!white,opacity=0.50] (0,0) circle (1cm);

end{tikzpicture}

end{document}

Here is my attempt with the torus:

documentclass[tikz,border=3.14mm]{standalone}

begin{document}

begin{tikzpicture}

%Torus

draw (0,0) ellipse (1.6 and .9);

shade[ball color = blue!10!white, opacity=0.5] (0,0) ellipse (1.6 and .9);

%Hole

begin{scope}[scale=.8]

clip (0,1.3) circle (1.55);

fill[white] (0,-1.27) circle (1.55);

end{scope}

begin{scope}[scale=.8]

path[rounded corners=24pt] (-.9,0)--(0,.6)--(.9,0) (-.9,0)--(0,-.56)--(.9,0);

draw[rounded corners=28pt] (-1.1,.1)--(0,-.6)--(1.1,.1);

draw[rounded corners=24pt] (-.9,0)--(0,.6)--(.9,0);

end{scope}

%Cut

draw[dashed] (0,-.9) arc (270:90:.2 and .365);

draw (0,-.9) arc (-90:90:.2 and .365);

end{tikzpicture}

end{document}

This is naive, since I'm using the same shading as for a ball, and I'm using clipping hacks to remove the shading from the hole. My attempt looks bad because it does not accurately reflect how light would hit the surface of a torus. Any tips for producing a realistically shaded torus would be greatly appreciated.

That's not a too serious answer, but just to tell you that if you have an idea how the shading should look like, you can reverse engineer it with pgfplots. Here is an example.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{pgfplots}

pgfplotsset{compat=1.15}

begin{document}

begin{tikzpicture}

    begin{axis}[colormap/blackwhite,

       view={30}{60},axis lines=none

       ]

       addplot3[surf,shader=interp,

       samples=60, point meta=x+3*z*z-0.25*y,

       domain=0:2*pi,y domain=0:2*pi,

       z buffer=sort]

       ({(2+cos(deg(x)))*cos(deg(y))}, 

        {(2+cos(deg(x)))*sin(deg(y))}, 

        {sin(deg(x))});

   end{axis}

end{tikzpicture}

end{document}

The trick is to play with point meta. Is this shading realistic? Certainly not (unless you have some crazy light sources). Could you make it realistic? Yes, if you know what you're doing or study the asymptote manual long enough. So if you want something realistic, use asymptote. If you want a cartoon and like to play with pgfplots, you may potentially find this useful.

Not an answer just an explanation of a setting or two available in @marmot 's excellent answer that may answer your question of translucency to an acceptable level (marmots that is) so if we take the above answer and tweak the one line (I found 50 to be a safer memory value than 60) the Opacity looks better around 7.5 where the far walls are ghosted through the surface materials, personally think this is more understandable than covering with broken lines see Drawing Torus with semi-dashed line on it

   addplot3[surf,opacity=0.7,

   samples=50, point meta=x+3*z*z-0.25*y,