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Is this for HEAT Service and Support (helpdesk/classic) or for the newer ISM platform?
Echoing Alasdair, what Product line is this for HEAT Classic 9.x versions or ISM 201X versions?
we are using heat 2016.2 so ISM
I'll move this to the ISM discussion board to help you in getting your answer then.
What we've done with our Tasks is make the Status and Owner fields read-only when the task is accepted. You could do the same for when it is assigned.
Put similar read-only/visibility rules on any quick action buttons available to interact with tasks.