You can use the networkdays function. It calculates the number of working days between two dates and you can also add holidays as additional arguments so the holidays are considered non-working days too.
networkdays(TASKOPENDATE, TASKCLOSEDDATE, holiday1, holiday2, .... etc)
Replace holiday1, holiday2 etc with valid numeric dates of the holidays.
No, I don't. I don't see why it's not feasible to create a list with the dates of holidays and load that into qlikview. Forget the saturdays and sundays. The networkdays function already considers those. You only need a list with the dates of the holidays. Surely your organisation has such a list somewhere, in an excel file or a database. Load that list into Qlikview.
// load the holidays in a temporary table as a comma separated list of values
Load concat(HolidayDate, ',') as HolidayList from ...somewhere...;
// create a variable with the list of holidays to use in networkdays
LET vHolidayList = peek('HolidayList');
// load the tasks and calculate the net working days per task
networkdays(TASKOPENDATE, TASKCLOSEDDATE, $(vHolidayList)) as NETDAYS