This is probably more of an Active Directory issue than a SQL Server issue. If your network team is changing DNS servers (adding or removing) those settings should be getting pushed to all servers (not only SQL Server) via a GPO.
As for the impact on SQL, there should be none, assuming DNS is working as expected, and properly replicating all entries to all DNS servers in the forest.
The only impact the RegisterAllProvidersIP setting has is whether SQL Server will register only the listener IP address of the local subnet of the primary replica, or if it will register all IP addressed assigned to the listener, for all subnets. If the SQL Server doesn't have a valid DNS server to talk to, it will not matter how many IP addresses it is trying to register.
In my experience, I've never needed to touch a SQL Server after a DNS change was made on the network. Now, it is helpful to be aware of such changes, so that should SQL Server all of the sudden become unable to registers its IPs in DNS, you'll have a good idea where the breakdown is.