Q: When upgrading a Hyper-V host from Windows 2008 R2 to Windows 2012, can I save the state of a running VM and resume it after the upgrade?

A: No, you can't save the state of a running virtual machine (VM). The save state format of Windows Server 2008 R2 is not compatible with Windows Server 2012, in the same way the save state format of Windows 2008 wasn't compatible with Windows Server 2008 R2. 

You should shut down all VMs on a host prior to performing an upgrade of the Hyper-V host. Note that if you have online snapshots of a VM which also uses saved state as part of the snapshot, then these snapshots will work after the upgrade to Server 2012.