Why my 2003 Chrysler Concorde limited battery keep dying

No, your starter will not cause your battery to die. If you verify all components are "good", then it may be necessary to do a "parasitic draw" test -- to see if an electrical component is staying "live" and drawing current when it shouldn't be, after the vehicle is shut off. Almost like a light bulb in the vehicle staying on (check this too), a stuck relay or electrical component can continue to drain the battery when the vehicle is off. Testing is needed.