No more than do the ICs that control the switching among inputs and outputs. In other words, no, within the context of the Rhea.
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I agree with Lew that doing what you describe is very unlikely to cause any negative sonic effects. It can be inferred from John Atkinson's output impedance measurements, described in the second paragraph here, that in contrast to some designs the Rhea's balanced and unbalanced outputs are driven from separate and independent output stages, so connecting one would not affect the other.
Without further knowledge of the specific design, BTW, I would NOT make the same assumption if you were to use both of the balanced outputs, or both of the unbalanced outputs, for your comparison. Especially, given the Rhea's high output impedances at some frequencies, if the input impedance of your integrated amp is relatively low, and/or the length of the cables is particularly long.
Also, keep in mind that when you switch between the balanced and unbalanced outputs you are changing several things at once. Besides changing the cables that are being compared you are changing the output circuit that is being used in the Rhea (the balanced and unbalanced output circuits apparently being considerably different designs, based on their very different impedance characteristics); you are changing the receiver circuit being used in the integrated amp; you are changing whatever susceptibility the two components may have to both ground loop issues and noise pickup (balanced interconnections generally having less susceptibility); and you are changing between fundamentally different operating principles of the two kinds of interfaces. So while the experiment may be useful in determining whether to choose between balanced and unbalanced interconnections in your particular setup, it would say little about the cables themselves.