I once read an article about proposed project to move a NEO (near earth object) using a small satellite that would land on its surface and drill out small pieces of the object, slinging it out into space. This force would "eventually" alter its orbit (trajectory). Is there any possibility that this could happen to Earth, as we continue to launch rockets out into space?
In principle, yes, but practically, no. For one thing, a NEO is much smaller than Earth, so the amount of effort required to slingshot the necessary mass is within reason (whereas the relative amount of mass of rockets being launched from Earth is infinitesimal).
This is also assuming that rockets are "slung" into space, but they aren't. The majority of a rocket's thrust is due to the exchange of momentum between the rocket and the exhaust -- not reactions between the rocket and the Earth (although you could say there is some due to atmospheric effects).
Finally, even if you were slinging mass from the Earth (or if the reaction forces from conventional rockets were significant enough), you would have to make sure each launch is timed correctly so that momentum is exchanged in the same direction -- otherwise the launches will generally cancel out.