Unlike other operating systems, kernel on Linux operating systems are slightly different. Kernel has a control on the functionality of drivers and hardware, whereas the applications and services are independent. It simply means that you can upgrade other part of your machine, without changing the exising kernel.
Probably, most of the system administrators would never like to change the existing kernel on their dedicated server, even they wants to upgrade other applications and services. Keeping the same kernel guarantees stability and continuous service to your clients and users. However, when an update of a security patch of your Linux distribution is unveiled it is recommended to upgrade your kernel as soon as possible or it may create a security hole for hackers and attackers.
Most of the Linux distributions allows the latest kernel version to be installed next to the old. This is beneficial, if your new kernel is not working fine, you can boot into the old kernel easily. Ensuring the correct kernel will help you to get rid of troubles. You must match your existing kernel type with the upgrade and then proceed.
Whenever a kernel is updated, a server reboot is necessary for things to take effect. Rebooting the Linux server in day time, when the traffic is higher, may have adverse effect on your traffic stats as well as running softwares. In this case, it is advisable to reboot your dedicated server only in the low usage times (ie. when the traffic is low on your website).
Lastly, you need to understand that, any customized kernel modules, especially those which are required for virtualization, will need to customization after the upgrade is done. However, some of the applications has the ability to compile the modules automatically to match with the new updated kernel.