As it mentions in the question line, is it possible to lock a task from being edited once it has it's resolution marked as done, or sits within the 'final' status?
Additionally, if it is possible, can we make it so that there is differing permission schemes for that, i.e: any user can't do it, but an administrator could?
You can apply a "jira.issue.editable = false" workflow property on status in the workflow. It's not a permission scheme, but you can also say "but group X still can"
Just to complement Nic's correct information, if you do not know how to set a property follow these steps:
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