I just tried this in a lab running on a FC connected array (vSphere 6). When you rename the volume you will get this message:
As per the message it will take the volume offline, rename it, then bring it back online. This process is very quick (split second) and so the VMs/ESX hosts may survive this short timeout. If you want to be extra sure though I'd recommend planning a maintenance window if you can take downtime on the VMs.
Once the volume is renamed the next replication update will update the downstream volume to the same name.
The renaming of the datastore doesn't cause any downtime.
It worked in my lab, but as above you may want to test first or plan a maintenance window to guarantee no impact.