Power a 5.5 V/1.8 V microcontroller from a lithium polymer battery: 12 V + LDO = 3.3 V, or just pull from one cell?
If you use a linear regulator the number of cells makes no difference to run time, since the same current is being drawn anyway. Whether that current is drawn from 1 or 3 cells doesn't matter, as the pack is deemed fully discharged when any cell reaches minimum voltage. 

However if you tap into a single cell then it will be discharged more than the others, so the battery will have to be balanced on recharge. The worst case scenario is if your ATTiny circuit is the only load, as then the other 2 cells in the pack will remain fully charged and cannot accept any current on recharge. 

Whether your charger can correct the imbalance depends on how it works and what other current is drawn from the pack. Some charge each cell independently, so it doesn't matter how unbalanced the pack is. Others charge the whole pack in series and bypass current around any cells with higher voltage, so to avoid overcharging the higher voltage cells the charge rate must not be be higher than the balancing current. 

If you have another circuit connected to the whole pack which draws a lot more than 0.33mA on average then the battery will be discharged long before it has time to go badly out of balance. In this case you don't have to worry about regulator efficiency, because its power drain is a small proportion of the total. Therefore it would be best to power the regulator from all cells, since that would avoid unbalancing the pack. With efficiency not being important and 7.4V or 11.1V available, you don't need an LDO regulator (a standard 78Lxx type would work fine).  

If reducing that 0.33mA draw would significantly improve overall efficiency then you are risking serous imbalance and probably should avoid tapping into one cell. In that case a switching regulator could improve efficiency. To do so it would have need to have a very low quiescent current draw (much less then 0.33mA) and high efficiency at 10mA.