While the Checklist API is Read Only, each returned checklist response contains a list of tasks and the list of answers for the checklist.
If you want to use the Checklist API to 'filter' by date, you can use the order_by function of the API to sort by date. Your API call would be structured along these lines:
Or, you could order by the lastupdated >
To sort in descending order you would put a - before the field so order_by=-closeddate