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I was wondering if general relativistic effects of the sun's rotation have also been measured, like gravity probes A and B measured GR effects from the earth.

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Great question! One of the earliest papers (to my knowledge) proposing such a test was by Haas and Ross via the Lens-Thirring effect. Aside: note that the 'gravitomagnetic' field doesn't actually involve magnetism per se.

Lorenzo Iorio reports a possible marginal detection of this effect. I'm not wise enough to comment on its validity/credibility, but I can say that many in the field are skeptical. This additional article might also be helpful.

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My understanding from hearing people talk about gravity probe B is that the Earth is actually easier (even with the distance/gravity well stuff aside), due to the noise generated by the solar atmosphere and magnetic fields. – Jerry Schirmer Mar 29 '13 at 14:16
Back in the 70's there was a flurry of activity involving tests of GR versus Brans-Dicke gravity. The oblateness of the sun was poorly known at the time, and if you took some measurements of oblateness seriously, it made Mercury's orbit violate GR. In general, the earth's structure, mechanics, and motion are much better understood than those of the sun. – Ben Crowell Mar 29 '13 at 20:16

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