Zinc is a reactive metal and lies above hydrogen in the reactivity series. So it displaces hydrogen from dilute hydrochloric acid. The reaction occurs as follows:
Zn+2HCl→ZnCl2+H2
Copper is less reactive and lies below hydrogen in the reactivity series. So it does not displaces hydrogen from dilute hydrochloric acid. Hence, there won't be any reaction taking place between copper and dilute hydrochloric acid.