Denefitly not! The reading glasses don't damage your eyes — they just improve how well you see. Also they can adjust your vision and improve the deading experience!
I think that non-prescription reading glasses that are either too weak or too strong for your eyes could be bothersome, but long-term wearing may bring effects.
Since wearing reading glasses will not cause your vision to deteriorate, it can bring better vision for you. So it won't damage your eyes, but you should care for eyes in daily life.