What a compelling idea! Yes, in fact it can.
Orbits are determined by a few factors and one of the largest of these are the orbiting bodys' mass. Removing mass from this object through mining, and then sending the material into space would have an impact on the distance of the orbit.
Not only that, but changing the orbiting mass will also translate into effects on the ORBITED body as well - since an orbit is characterized as two separate objects orbiting a third "balance" point.
O -x------ o <------- this is a crude example of what I'm talking about. The small o is the moon and the O is a planet. They orbit around the x which is not how many people think of orbits- instead assuming one rotates around the other.