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For a given black body radiation curve, would the changes to the spectrum resulting from cosmological expansion and those from Doppler effects be distinguishable on the basis of the shapes of the resulting curves alone? Or, put another way, starting from the same spectrum, can both processes produce the same observation (for suitably chosen magnitudes of expansion or velocity)?

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The redshift due to cosmological expansion is identical to a Doppler shift in its effect on the spectrum of any source. To be specific, both phenomena "stretch" all wavelengths by the same factor.

There's a very good reason for this: in a suitable coordinate system, the cosmological redshift is a Doppler shift.

You'll find statements in some textbooks saying that this isn't true, but a weak version of this statement, which is nonetheless strong enough to explain why the effects on the spectra are identical, is uncontroversially true. To be specific, the redshift of a distant galaxy can be thought of as the accumulation of many infinitesimal Doppler shifts along the line of sight. (Each member of a family of comoving observers is in motion relative to her neighbor, and each can "watch" the redshift build up gradually due to these relative velocities.)

One perspective on this subject (mine, to be precise) can be found in this paper. Even if you don't like our point of view in this paper, our description of and references to other treatments may be of interest.

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One caveat: they may be different if the size of the object approaches the hubble radius, in which case, $\frac{{\dot a}(t)}{a}$ will have a different value for light coming from the near side of the object than it will for light coming from the far side of the object, so you should get a broadening of a spectrum relative to the doppler shift of a rigid object moving with a uniform velocity. Of course, no known objects are anywhere near this large. –  Jerry Schirmer Mar 24 '11 at 18:01
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Hmm, maybe, I guess, although it's confusing to think about such large objects. If all parts of the object are at rest in comoving coordinates, then the object itself is expanding -- i.e., different parts of it are moving relative to each other. A Doppler picture is still OK: you see a superposition of Doppler effects corresponding to different speeds. If the object is rigid, then things are more complicated. For one thing, it's not entirely clear what coordinate-independent notion of "rigid" to apply in this case, since the object doesn't live in a single inertial frame. –  Ted Bunn Mar 24 '11 at 18:12
    
So what additional line of evidence is used to conclude that space has expanded since light was emitted from an object, rather that concluding that the object was just receding from us when the light was emitted? –  raxacoricofallapatorius Mar 24 '11 at 22:01
    
And if "stretching" were caused by doppler shift rather than cosmological expansion, wouldn't we (Earth) have to have some privileged position, since everything would have to be rushing away from us? –  raxacoricofallapatorius Apr 22 '11 at 17:17
    
There's no need for a privileged position or reference frame. If you want to think of the redshift as a Doppler shift, it follows that everything is moving relative to us. But we're also moving relative to all of them. To answer your previous question, I claim that there is no line of evidence to show that "space has expanded" as distinct from "the object is receding." My claim is that those statements are equivalent descriptions of the same physics, just using different coordinate systems. There is no "fact of the matter" about which one is true. The paper I cited argues for this conclusion. –  Ted Bunn Apr 22 '11 at 17:37
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This addresses the comment to Professor Bunn's answer more than the original question:

Any spacetime described by the FLRW metric, where objects are static in comoving coordinates and redshift is due to expansion of space, can in principle be also described, via an appropriate change of coordinates, by a spherically-symmetric (SS) metric, where objects move along radial timelike geodesics and redshift is due to positional (gtt) and Doppler factors. Since both metrics describe the same physical system, they are observationally equivalent in all respects.

To note, the equivalent SS metric is static, meaning that gtt and grr do not depend on time, only in the empty (Milne) and lambda-vacuum (de Sitter) cases. Moreover, even in the next simplest case, the flat matter-only FLRW model known as Einstein-de Sitter model, gtt and grr cannot be expressed algebraically in terms of non-comoving time, as I mentioned in my answer to this question. I take it as a strong hint that the FLRW model with expansion of space is the real thing, but even if you take a purely utilitarian position, you can at least say that expansion of space is the only conceptual framework with which it is possible to work.

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For what it's worth, I agree with everything except "I take it as a strong hint that the FLRW model with expansion of space is the real thing." As far as I'm concerned, the fact that different coordinate systems lead to different descriptions of the phenomena mean that there is no "fact of the matter" about whether space is "really" expanding. I agree that expansion of space is a useful model within its limits, but I think that those limits are not always recognized. –  Ted Bunn Mar 25 '11 at 14:13
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