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If you make a context mode perspective that is grouped by due date, you can close the groups corresponding to due dates in the past and save the perspective with the restore expansion box checked. It's not quite an ideal situation because you have to decide whether to keep today's tasks visible or not, and you can have a mix of overdue and not-yet-overdue tasks in the today group.

A different approach you could use is to go into the OmniFocus Style preferences, select the style for Overdue, and change it to make the text invisible. There will still be a checkbox visible, and such actions will still occupy space, but anyone using Forster's methods needs to be good at picking things out of a page full of clutter, right? :-)