Question

As the earth loses mass because of burning fossil fuel, has the earth moved closer to the sun?

No, the earth does not lose mass, and thus the earth is not falling towards the sun because of the burning of fossil fuels. When fuels are burned, the carbon dioxide that results still has mass and is still on the Earth--it is just in the atmosphere rather than on the ground. So the burning of fossil fuels will not result in any change in the orbit of the Earth. Of note, however, the moon is slowly moving away from the Earth, a few millimeters per year. This is because the tidal bulges on the Earth (largely in the oceans), raised by the Moon are rotated ahead of the Earth-Moon line by the Earth's rotation since it is faster than the Moon's orbital motion. The gravity from these leading and trailing bulges impels the Moon mostly forward along the direction of its motion in orbit (the Moon's orbit is not exactly in the plane of the Earth's equator). This force transfers momentum from the rotating Earth to the revolving Moon, simultaneously dragging the Earth and accelerating the Moon. In fact, the Earth’s day is slowing down at a rate of .002 seconds every century due to these forces!
March 11, 2002