I assume this is a question about Moteino more than about battery chemistry and complete discharge effects.
RE depleted batteries - just recharge them when you get a chance. If you plan to be away a very long time after they are dead, then increase capacity - what use is a moteino that does nothing?
If your batteries are built and used properly, they will not overheat or anything like that.
Depending if the BOD is active or not, the moteino will stop working when entering the BOD threshold. From other tests performed by myself and others, if you disable BOD, then the atmega328p will stop working around 1.6v (moteino without the LDO, run at 8mhz). LiPo cells are considered depleted around 3V.