Not a direct solution, and maybe not the best approach, but here's an idea to achieve similar result (I once had similar requirement).

Create a field of function type (fields.function) which check several values in the account.invoice object (and another related objects such as payment -> account.move.line) and compare these values:

• state
• date_due
• account_move_line.date (many2many)

Beware that this would probably slows down tree view a bit.

Another idea and probably a better approach is to write the paid_date to account.invoice when the state changes from 'open' to 'paid'. I don't know but it may also be possible to check the wkf_item object checking it's workflow transformation date, could someone confirm this?