Generally speaking, every zone is different and most are operated by different companies. Even if it's operated by the same company, they usually operate separately, and therefore if you change the lot/location/zone you would need to repay.
The one exception to this is if its a City that allows spot hopping (Eg City of London). In those cases you don't need to pay again you just move your vehicle to the new zone within the downtown core where you paid.
However, most Cities require repayment when switching zones.