There is no point arguing this. One inertial frame of reference cannot travel at the speed of light relative to another. So if the earth is accelerating for an infinite amount of time, it will approach the speed of light relative to another inertial frame of reference regardless of which in the universe you choose. Thus earth will become increasingly heavier and would have to be propelled by a force infinitely powerful.
So the question is, what constitutes the force, and more importantly, what fuels the source?