Is putting pots on the stove without oil or water damaging the pot?
Assuming your mom was talking about putting them on the stove while the burner was on, yes, she was right.

It is ok to warm them slightly just prior to adding a little oil. Some people say it helps the subsequent fried foods not to stick. Don't know if that is true, but don't leave the room and let the empty pot get too hot, which can happen very quickly. 

Overheating can cause thin pots and pans to warp. An enamel coating on a pan can chip off, or even melt onto your heating element. "Teflon" type non-stick coatings can burn off and be damaged releasing toxic gases into the air. Even cast iron can get too hot, and the seasoning can be messed up.