[Rancher] As part of a Rancher upgrade, do I need to upgrade my k8s cluster to a version listed in the support matrix for that Rancher version? If I do not, will the cluster continue to work fine as is? Will Rancher support this cluster?

It is recommended to move to the k8s versions listed in the matrix for a Rancher version, but not doing that should not break things. The most important is to stay within the Support Matrix. 
If you have a cluster running Kubernetes version 1.25, for example with Rancher version 2.7.x, and you'd like to go to, for instance, Rancher 2.8.2, you can stay at 1.25 until the Rancher version upgrade is completed. Rancher Support will continue to help should there be a need. 

That said, should there be an issue on that cluster that requires a fix in k8s v1.25, we may not be able to do that and would, at that point, require an upgrade to a higher-supported k8s version (for example, if a fix is available in 1.26). 
We always suggest to our customers to stick to the support matrix and run higher versions where possible - especially because of security fixes and other crucial features.