One sees immediately from (3.32) that no drift occurs when H(z) = H0 (1+z). From the definitions of H and 1+z in terms of a(t) it is easy to find the solution: \dot a = constant or a=t/t0. This is the mark of a Milne, or empty (Omega=0), universe [see Think About 3.1].
This result makes the role of the Principle of Equivalence explicit: lack of acceleration (\ddot a = 0) is lack of gravitation. A Milne universe is mathematically equivalent to Minkowski spacetime; there is zero spacetime curvature and of course Minkowski spacetime is that of special relativity - no gravitation. Also see S3.5.3 and the discussion of Equation (3.33) relating redshift drift to acceleration just as redshift itself is related to velocity (e.g. Doppler shift).
So the redshift shift test is measuring the acceleration of the universe. This is particularly clear from the low redshift expansion of (3.32) where one sees the effect is proportional to q0.
Note that a Steady State universe, which one naively thinks of as having no time dependent phenomena, does in fact have a redshift drift. In this case, H = constant, and d\ln z/dt = H = constant or Delta z/z = H Delta t. This is the same as taking q = -1.
A further note on the possibility of observing this effect: even if one could make extremely stable and precise redshift measurements of a distant object over a long time span there are other factors entering. The expression (3.32) is valid for a quiet universe, where the time dependences are dominated by the Friedmann expansion. There are several other effects in our universe that increase the difficulty of such a measurement, just like the caveats in S3.5.3 on the period test. Any gravitational acceleration will enter as well. One example is the Rees-Sciama effect [S8.5.3, p180] of changing gravitational potentials along the line of sight. Another is the Sachs-Wolfe effect [S8.5.3, p180] due to passing gravitational waves. In order to detect the drift over a 100 year baseline (z = 10-8) one would need the energy density in gravitational waves to be OmegaGW < 10-16. But the energy density in kpc to Mpc wavelengths is not that well constrained.