I thought I'd follow up with my solution to this bug in case anyone else is interested. It is based on a really excellent StackExchange response to a similar question here: https://mathematica.stackexchange.com/questions/8865/pass-function-or-formula-as-function-parameter

Here is the code that I wrote based on that. Please note that I've only been using Mathematica a few days so it is entirely possible/likely that I've made some stupid mistakes. But it seems to work:

QPlot[fn_, intvl_, options___] := 
  Module[{func, externalSymbol, localVar, minQ, maxQ, unit}, 
   externalSymbol = Unevaluated[intvl] /. {x_, y__} :> HoldPattern[x];
   {minQ, maxQ} = Rest[intvl];
   unit = QuantityUnit[minQ];
   func = 
    Unevaluated[fn] /. 
     externalSymbol :> Unevaluated[Quantity[localVar, unit]];
   Plot[func, {localVar, QuantityMagnitude[minQ, unit], 
     QuantityMagnitude[maxQ, unit]}, options]];

In short, the idea is to strip the units off of the range limits and then put them back in before calling the function.

Thanks for another quick response. I wasn't familiar with the ListLinePlot function so it is good that I can use that to work around this bug.

I don't understand your comment about Plot "understanding" that the x-range is from 0 to 90 degrees. It isn't plotting it that way -- whether I choose to use:

Plot[Sin[x],{x,Quantity[0,"AngularDegrees"],Quantity[360,"AngularDegrees"]}]

or

Plot[Sin[Quantity[x,"AngularDegrees"]],{x,0,360}]

they should clearly yield the same result and they don't.

Here is another example which simply doesn't plot at all (even though you can evaluate the function, e.g. using your ListLinePlot, and it works)

f[x_] := x/Quantity[1, "Inches"];
l0 = Quantity[0, "Feet"];
l1 = Quantity[1, "Feet"];
Plot[f[x], {x, l0, l1}]

So I'm pretty sure that Mathematica has a bug that it doesn't understand Quantities (treat them properly) in the notation {x,xstart,xend} in the Plot function. Though, again, if I'm missing something important here please let me know.

Thanks for the quick response. The first plot is wrong, it should look exactly like the second one. I see the same output that you do.