No, if you renew your license for the same number of devices, any days remaining on your current license will be added on to your new one.
However, if you change the number of devices, your new license term will only begin once your current one has ended. Add your new activation code to your Kaspersky product as normal and your new license will be activated automatically when your current license expires.
If you are having technical issues renewing your Kaspersky product, take a look at the Activation Errors section of the Knowledge Base. We’ve written up solutions for the most common renewal and activation issues.