How does taking the absolute value of a complex signal reflect in the frequency domain?
Absolute value is not a linear operation, so no, there is not a consistent dual in the frequency plane.

Having said that, though, it is somewhat similar to calculating the signal power, which involves multiplying $x[n]$ with it's complex conjugate.  Multiplication in the time domain is equivalent to convolution in the frequency domain.  The result in the time domain is real, so the result in the frequency domain is even (i.e. symmetric around 0 Hz).

Absolute value will probably produce a similar effect to calculating the power since it is the square root of the power.  It is also real so it will be even in the frequency plane.