GitHub introduced some time ago Task lists.
What are the minimum permission I need to set-up so collaborators/assignees are able to tick those boxes on the issue of the project?
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You need to give write access to your repository, as checking off a task list item in an existing comment is an edit action.
This is the explanation which I've received from GH support:
We have discussed the possibility of offering a more refined permission scheme. There, people will be able to edit issues, but not push to that repository. I am afraid I don't have anything concrete I can promise currently though. Our team will consider it.
On the wiki, it may be that collaborators are by default able to check of the boxes on issues/pull requests, since they can edit them.