Designing an over discharge lithium ion protection circuit
Some such circuits DO keep slowly discharging the 'protecte' cell.

It is possible to design circuits which "self disable" so that they draw no current once they have performed their function. [I designed portable solar lights which did this. The circuit was re-enabled by a feed from the (solar) charging circuit.

Current discharge can be extremely low if desired while still retaining a testing function. eg at say 1 uA drain you lose 1 mAh every 1000 hours or about 40 days.
Custom monitoring ICs could be implemented with drains of well under 1 uA. 

You could use eg a zener diode in the monitoring circuit so that when the voltage falls below about Vzener (essentially) no current is drawn. As zeners are of low precision this could be used to disable the higher precision monitor.