Cheap reading glasses won’t damage your eyes, but they may decrease the amount of time you can read comfortably, depending on your prescription.
Inexpensive reading glasses typically have the same magnification power in each lens. This is beneficial to people who have the same prescription in each eye, though most people do not and are thus best accommodated by prescription lenses. If one doesnt have the same prescription in each eye and wears glasses with identical prescriptions, one eye will be under corrected/over corrected. This will cause fatigue in said eye, which will, in turn, cause discomfort after a long period of use.