I believe it will still work even though you will get errors from CDP or STP.
STP would actually partition, meaning you would end up with two seperate STP domains.
CDP may still work but VLAN 60 would have to exist through out your environment, regardless of whether it was tagged or running as native. Once CDP encountered a boundary where VLAN 60 moved from native to tagged you would get the errors Soulja mentions above.
The place you are going to have the biggest issues beyond STP would be any trunks between switches with different native vlans. The trunks should come up but they are going to be unstable and problematic.
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