Depending on how the mass is removed, the trajectory can change substantially or not at all. It all depends on whether the removal process applies any kind of force to the asteroid. So if a mass driver sitting on the asteroid surface is used to throw mined material off the asteroid in one direction, then conservation of momentum (mass x velocity) requires a net force on the asteroid in the opposite direction (3rd law of motion). if the mass driver is on the surface and pointing perpendicular to the radius vector to the center of mass of the asteroid,, it will actually provide a torque = mass x velocity x radius that will change the angular momentum of the asteroid causing it to spin either fast or slower (or in a slightly different direction). On the other hand if the material is mined and then brought to a rocket ship that takes off from the surface, there is no net force on the asteroid, except for some gas pressure from the rocket exhaust while it is still close to the asteroid.

Moving mass from the asteroid will give it a boost, just like a rocket engine. The mass driver is an option to alter the trajectory of asteroid in the NEO hazard mitigation. In my opinion in the asteroid mining the trajectory change needs to be calculated since the altered trajectory may impact the Earth or the Moon. This is a interesting topic, and deserves attention.

well, I guess for shifting the orbit of an asteroid we need to concentrate on ΔV. We can deflect it's trajectory  by using space tug method and can shift it to the desired location (may be LEO, where the mining unit can be placed). Shifting of an orbit will be safe  if it is slow.