Olaf has already explained the required steps to remove the VLAN, however I thought to give a bit of the background information on EAPS here and what you would want to do..
If you have a VLAN on all 4 switches of your drawing on all ring-ports, then you create a loop. The concepts of Ethernet switching indicates that traffic is being sent out all ports on the same VLAN with the exception of the receiving port. So let's say Core 1 receives a packet on port 10:1 and it does not have an FDB entry for it (to know which port to forward the traffic to), it will do "unknown unicast flooding", so it will send the traffic out all ports with exception of the incoming port.
In this example it will send the traffic out port 10:2. Core 2 receives the packet on port 10:2 and will do the same, so it will send it out of port 10:1 towards Core 4 in your drawing. The same happens again on Core 4 and Core 3 and eventually the packet is being received by Core 1 again, only to start the process over and again forward it out to Core 2 and this is how a loop is created.
To prevent this loop from happening you can use EAPS. The EAPS master switch blocks the secondary port on the protected VLANs, so that this loop is being interrupted.
This is similar to just removing one of the ports from the VLAN, however if you actually did this, and there was a single link-failure in the network you would split the VLAN into multiple segments. To avoid this, you can i.e. use EAPS, which simply "opens" up the blocked secondary port and full connectivity is restored.
So when you say that a specific VLAN is not required by EAPS anymore this indicates that it's not part of the full ring (so the VLAN has been removed from at least one ring-port of one of the switches).
In this case you can just delete it from the list of protected VLANs, with the command that Olaf had already mentioned as there is no chance that a loop can be created.
This does not mean, that you will have to remove this VLAN from all ring ports on all switches, as i.e. in your case you still need connectivity to some servers.
Just make sure that before removing this VLAN from the list of protected VLANs in EAPS it is not creating a full loop anymore (so have at least one ring-port removed).
