Is the moon's orbit going to be changed significantly enough due to its mass reduction
Mass reduction will have zero effect on the moon's (Titan's) orbit. Mass of the moon is not part of the orbital calculations (mass of the planet is). It is possible that the method of removing mass might shift the orbit, but since you don't give the method, we can't really describe that.
It will affect other orbits, but mostly by affecting them less. Currently Titan distorts other orbits with its mass. Removing mass will decrease this effect, but it's not that big an effect in the first place. This will not cause observable changes in the time scale of a typical story. It might cause noticeable changes if your story is lasting millions or billions of years.
Changes to Titan's mass would have minimal effects on the rest of the solar system.
If your goal is to exhibit changes, consider having the aliens harvesting something from Titan's core rather than its atmosphere and seas. Chunks of Titan could become asteroids and comets, flying all over the solar system and bumping into things. Or have them pull Titan out of its orbit while they extract what they need, letting it go when done. Either could give you a Deep Impact or Armageddon style story (asteroid impact with Earth).
Newton's law of universal gravitation says that the force with which two objects are pulled together is proportional to the product of their masses. However, that force is not part of orbital mechanics. What matters in terms of orbits is the acceleration.
Essentially what happens is the mass of the satellite is divided out of the force equation to give just the acceleration. It's combining two equations:
- $g$ or $a$ is the acceleration due to gravity.
- $G$ is the gravitational constant.
- $m$ and $m_1$ both represent the mass of the satellite/moon.
- $m_2$ and $m_p$ represent the mass of the planet.
- $r$ is the orbital radius, the distance between the centers of mass of the two objects.
An orbit exists when the acceleration due to gravity balances the inertia of the tangential velocity in such a way as to maintain the tangential velocity's magnitude even while changing its direction. Neither the acceleration nor the velocity is dependent on the mass of the satellite.
The mass of the satellite does affect how much the planet revolves around the satellite. But in a typical planet/satellite relationship, the masses are so different that we ignore the effect of the planet revolving around the satellite. Reducing the mass of the satellite essentially makes the planet wobble a little less.
There is a slight effect on the satellite's orbit from its mass. When the satellite moves the planet, it changes the orbital radius. That in turn changes the acceleration equation. But the change will generally cancel out over the course of an orbit.
If Titan were larger, increasing the mass might pull Saturn out of its orbit around the Sun. But decreasing the already small mass is unlikely to have a large effect on Saturn. Even at the worst possible time, it's unlikely to be a big deal. I'd actually be more worried that the alien spacecraft, which is apparently huge (large enough to carry off a substantial portion of Titan's mass), would disrupt Saturn's orbit.