The add method accepts an optional IsChecked parameter. Then you can add your objects to the marked list box in the correct state.
List<Tasks> tasks = db.GetAllTasks();
foreach (var t in tasks)
tasksCheckedListBox.Items.Add(t.Name, isChecked);
Or you can change the checked state of an element after adding it something like this:
foreach(var task in tasks)
{
tasksCheckedListBox.SetItemChecked(clb.Items.IndexOf(task), isChecked);
}