This is normal. When you initiate VMware Tools install/update, vSphere will connect given ISO file to your guest's virtual CD drive. The ISO file resides on your ESXi host.
I don't think you need to unmount it from your guest manually, it is one of the final steps of the install / update process (you should see it in the console when the operation finishes).
Unmounting the drive in the guest will not disconnect the ISO though. In order to do this, you have to go to your vSphere client, select your VM and click Guest -> End VMware Tools install in the Actions menu. This should disconnect the ISO. If it doesn't (which happens a lot) simply edit your VM's settings and change the Device Type under your CD/DVD drive settings back to Client Device.
This can be slighly irritating, especially when you need to evacuate an ESXi host quickly and you have to edit several VMs first. Rest assured that this happens to all of us :-) Consider using the open-vm-tools package instead. In RHEL 5 you might need to add a new repo for this, see https://plone.lucidsolutions.co.nz/linux/centos/vmware-tools-rpm-repository-for-centos-5
I typically unmount the iso once I have moved the contents locally to be installed.
On the two that are having issues, I'm not able to click Guest > Install/Upgrade VMWare Tools, it is greyed out.
If I select the radio button Client Device from Datastore ISO file, it allows me to do that and the entry under the field under Datastore ISO File disappears. And we can vMotion with no issue now too. Not sure why before it gets stuck.
I'm not familiar with open-vm-tools package. Are there differences between what that is and what is offered by VMWare Tools? Are their pros and cons?
There is a KB article on this: VMware Knowledge Base
I think even VMware nowadays recommends using open-vm-tools. The main advantage for me is easier maintenance. OVT updates automatically whenever I trigger yum update. I do not have to update it separately.
If I were to start using open-vm-tools instead of VMWareTools, will the guest VMWare Tools status reflect current version under the General area?