Well, there have been many experiments with the earth and sun. It has been proven that the earth tracts to the actual location of the sun, or about 2 degrees ahead of the apparent location that we see as the sun. I remember reading on some website a few years ago that if there were any significant lag in the planetary reaction, that the planets would spiral out of their locations. Think about it! Have some ball orbit a center point and then move the force acting on the ball so that it is pointing a few degrees off the center point. what happens to the path of the ball?
I am familiar with the works of Van Flandern and his finite speed gravity inducing orbital decay. I've not done the calculations myself nor verified the work but his arguments are compelling. Either way, Gravity Probe "B" should provide some conclusive data and it will be very interesting to see the results.