First, assuming that this a straight-on shot, anything else is way too complicated to figure out. So, you have a relativistic moon ramming into the Sun. Even within just the extreme outer layers of the Sun enough energy would have been liberated from the kinetic energy of the collision to completely vaporize the moon, and from then onward you have a high speed jet of dense plasma shooting toward the heart of the Sun. I'm not sure whether it will get there, but I am sure that it will create a new "core" in the Sun. A region of compressed, high density, high temperature plasma. The amount of energy behind this process would be roughly 1/6th of the gravitational binding energy of the Sun, which is a tremendous amount indeed. This will basically cause a nova that will substantially disrupt the Sun, possibly even blowing it apart, depending on how much energy from fusion reactions is released. The Sun would basically be wrecked afterward, but I couldn't say for sure whether it would be brighter or dimmer (I would guess brighter). In any event almost certainly everything on one side of the Earth would be insta-fried, and the Earth's atmosphere would almost certainly have been stripped away, ending almost all life on the planet in minutes.
Any object larger than about 6x the size of the Moon would deliver more kinetic energy to the Sun in the impact than the Sun's gravitational binding energy. A lot of the same processes as above would play out, but almost certainly the Sun would be ripped apart, exploded into a kind of supernova. Earth would be utterly ruined and possibly even shattered into pieces or melted apart.Source