Cluster core resources comprise a cluster IP, cluster network name (associated with cluster IP), and a quorum resource.
Your SQL instances have their own set of resources which include both a cluster IP and cluster network names which are separate from the cluster IP and network name. They operate independently, so the fact that the cluster IP and network name are offline should have no effect on the operation of the SQL instance.
Cluster resources being offline would affect management of the cluster, as those resources are needed for management of the cluster. However, if the cluster resources remain offline for an extended period of time, the cluster will stop working. Being offline can be a temporary condition, as the cluster will try to bring the resources back online.