File Share Witness with 4 Node Cluster
No, the cluster will not go down until a node cannot form quorum.  Assuming all four nodes have a quorum vote, there are five possible votes.  Accordingly a node will have to get three votes to maintain or gain quorum.  When the file share witness goes offline, the node that currently owns quorum will maintain ownership because it can still get four votes.

See Understanding Quorum in a Failover Cluster for the long answer.