If the earth lost 1/81th of its mass how would it orbit change? If a planet becomes lighter, then I believe its orbit will change.

For example, the moon is 1/81 of the mass of the earth. So, if the moon disappeared, then the earth-moon system would lose 1/81th of its mass. How would the orbit of the earth change in that circumstance?
I will answer the specific case of the Moon disappearing since the more general question, is a bit open-ended. 

Of course the Moon cannot just "disappear." However, there are very real circumstances where a planet or other stellar object might be disrupted. The best example I can think of is if a sufficiently large asteroid or moon from another planet somehow make their way into Earth's orbit and collide with the Moon. Such collisions between Solar System objects are rare, but occasionally occur in models of the future evolution of the Solar System (see work by Jacques Laskar). 

However, one must be careful to first consider whether that mass is actually lost from the system. For instance, in the Giant Impact Scenario a Mars-sized object is theorized to have slammed into the Earth, but the material never left the system. If a similar collision were to disintegrate the Moon and the mass remained as rocks and dust in orbit around the Earth, the effect on Earth's orbit would be minor. This is because it is the Earth-Moon system that orbits around the Sun. Note that there is a small change since the Earth-Moon barycenter has shifted position, but the effect is tiny so let's ignore it for now.

Instead, let's consider a case where the collision disrupts the Moon, and its material leaves Earth's orbit. If this occurs sufficiently quickly, (faster than the orbital period, i.e., 1 year) then from an orbital dynamics standpoint, the mass is effectively lost instantaneously. The Dutch astrophysicist Adriaan Blaauw, first calculated the relevant physics. Earth's orbit will expand, but by an almost miniscule amount. The reason is that the dynamics of the Earth's orbit around the Sun is dictated by the total mass of the system (i.e. Earth's mass plus the Sun's mass). To lowest order, the new orbit's orbital separation ($a$) will be:

$$ a \approx 1 + \frac{\Delta M}{M} {\rm AU} \approx 1 + \frac{1/81 M_{\oplus}}{M_{\odot}} {\rm AU} \approx 1 + 4\times10^{-8} {\rm AU} $$

So, the orbit will expand by roughly 40 parts in a billion. A quick calculation using Kepler's third law ($a^3\sim P^2$) shows that the length of a year would increase by about 2 seconds. A similar change in the Earth's orbital eccentricity will occur ($\Delta e \sim \frac{\Delta M}{M}$), but since this change is smaller than the precision to which we have measured the Earth's orbital eccentricity, the effect is negligible.