Time relativity relating to Carbon dating?
Earth travels along its orbit at about 30km/s. If you plug that number into a time dilation calculator, you'll find that one second on Earth passes every 1.0000000050069 seconds in a Sun-centric reference frame.

Carbon-14 dating is good only to about 40,000 years into the past; Earth's time dilation would reduce that 40,000 years by a little under two hours, or far below the precision of the technique.

Edit: I'd like to address three more issues. First, the rotation of the Earth's surface only results in about 0.5km/s at the equator, and so is insignificant compared to the orbital velocity. Second, nothing has changed the Earth's orbit for many millions of years, probably for billions of years, so that would also have no effect on the dating.

Third, and more important, we're all traveling along on the Earth at (approximately) the same speed. So you, I, and a group of C-14 atoms in a fossil will all experience the same time dilation.