No!
Actually, when you purchase your upgrade or receive a new serial number by virtue of a maintenance upgrade, nothing happens until you want it to.
Whenever you’re ready to install the upgrade, you can do so on the same computer or on a different computer. When you install and register the license with the new serial number, a 60-day “clock” starts counting down on the older installation. When the 60-days expire, the old license will de-activate and will revert to running in trial mode.