Nice question. The gravitational effects of the pressure components have been directly verified in at least two ways that I know of. The early universe was radiation-dominated, so I don't think you can reproduce any of the relevant cosmological data (e.g., big bang nucleosynthesis) if you leave out the pressure terms. Kreuzer 1968 is a laboratory test, and Bartlett 1986 tested it using lunar laser ranging. I've written an exposition here of how Kreuzer can be interpreted as a test of pressure as a source, and this review article by Will summarizes the Bartlett experiment in section 3.7.3.
Others might be able to comment on direct experimental tests of the shear stress and momentum density as gravitational sources, but if they didn't contribute, you'd be breaking Lorentz invariance, since a diagonal stress-energy tensor gains off-diagonal elements under a boost.
The precession of Mercury's orbit can be explained in terms of the motion of a test particle in a Schwarzschild metric, so it doesn't test anything about the non-00 terms in the stress-energy tensor.
It seems likely to me that Gravity Probe B's detection of frame dragging can be interpreted as a direct test of the momentum-density part (since I think it can be interpreted as a gravitomagnetic effect).
Bartlett and van Buren, Phys. Rev. Lett. 57 (1986) 21.
Kreuzer, Phys. Rev. 169 (1968) 1007