No. The current versions of ArcGIS Server no longer require these ports to be available for an ArcGIS Server site.
In ArcGIS Server 10.6 and earlier, ports 4000, 4001, and 4002 were assigned for communication between ArcGIS Servers when creating or joining a site. If a cluster is created, a cluster port is specified to allow communication on all machines in the cluster. In a typical site, the default cluster uses port 4004.
However, due to performance issues with multiple-cluster sites, the use of these sites for new deployments is discouraged, and any multiple-cluster site deployment is advised to migrate to single-cluster mode since 10.4. Since ArcGIS Server 10.7, it is not possible to configure multiple-cluster sites, and these sites are reconfigured to a single-cluster architecture when upgraded to this version. Multiple ArcGIS Server machines can still be joined to a single site.
Therefore, ports 4000, 4001, and 4002, and 4004 (clustering) are no longer required for the current version of ArcGIS Server.