Yes because ions are... ions. You don't see photons knocking the planet out of orbit, do you? They can exert force but they don't exert that much force.
But if you're trying to avoid self-propulsion and want to launch from Earth, say, even a 200kg craft, anywhere "close to the speed of light", then that will most probably require a significant enough amount of force to knock the planet out of orbit.
If you can apply small force over a long time, that will get you up to speed, too.
Someone did the math in the thread, and suggested that a constant 1g of acceleration would get you to the centre of the galaxy in 20 years (as measured by the clocks traveling on your spaceships). 1g of acceleration for 200kg is about 1962 Newton.
(This back of the envelope calculation assumes you have eg someone fire a laser at your ship to give you the energy you need. If you need to bring your own fuel, the rocket equation increases the total mass needed. But the same principle still applies: something like an ion drive has very little force, even if the top speed it can reach can be enormous.)
I think everyone is collectively ignoring that I'm specifically talking about accelerating the craft from earth, not using an ion drive or similar form of propulsion on-board. As that's what is implied by the comment that I responded to: "If you could shoot it out at close to speed of light"
But if you're trying to avoid self-propulsion and want to launch from Earth, say, even a 200kg craft, anywhere "close to the speed of light", then that will most probably require a significant enough amount of force to knock the planet out of orbit.