Question

If the moon would suddenly disappear would the orbit of the earth around sun change in a way that we would notice?

Answer 1

The Earth and moon orbit their barycentre (located about 100km below the surface of earth). It is this barycentre that obits around the sun. SO if the moon were to suddenly disappear, the earth would no longer orbit the barycentre (instead it would orbit its own centre of gravity), and so would continue around the sun either a little closer, a little further, or unchanged, depending where the moon was in its orbit when it vanished. This would not have a noticeable effect to most people i think, but astronomers would certainly notice it as they monitor the skies in a far greater precision than others.

Answer 2

Just noticed a typo. The barycentre is 1000 km below the earth's surface, not 100 km. Forgot to hit the zero that one extra time.