Is there any benefit to using a voltage booster (such as Batteriser) on "dead" batteries?
"Is there any benefit to using a voltage booster on batteries that are below a device's cutoff/operating voltage?"
Of course there are benefits in that situation: a battery that would otherwise be dead can still be used for some time. But probably not for long, so it is debatable whether this is usefull.
What DJ (IMO correctly) argues is that the Batteroo claims are wildly exaggerated at best, and the use of their device with batteries that are not yet below the cutoff voltage will lead to some additional energy use, so the overall effect might be negative.
Our goal is to keep the load on the batteries running as long as possible. In general, these loads are either fixed resistance (like a basic flashlight) or fixed power (like almost anything electronic beyond a certain complexity). A fixed power load is generally a switching regulator, which has a minimum dropout voltage.
A fixed resistance load doesn't much care what the input voltage is; the power out of the batteries will drop with the square of the voltage. Your bulb gets dimmer as the batteries die, but the dim bulb consumes less energy. You get a little time running bright, and a long time running dim. By putting a boost converter on the batteries into a resistive load, you're effectively turning the lamp into a fixed power load. Now, the lamp runs bright until the dropout voltage is reached, at which point the lamp stops entirely.
If the load was already fixed power, adding another regulator in front of it doesn't change that. The only possible effect you can have is to change the dropout voltage. If you've made the dropout voltage higher than it already was, you've made the device run for a shorter time! If you've made the dropout voltage lower, then you should be able to run the same device until a lower voltage point on the batteries.
However, the total energy you get out of a battery by putting a fixed power load on it is very complex; at lower voltages, you necessarily draw more current, to make up the fixed power (P=VI). The more current you draw, the more the terminal voltage drops due to internal series resistance, the faster the battery dies, and the less total energy you get out of it. So you could only ever be increasing the total energy draw from the batteries by a very small amount, and that amount is almost certain to be consumed by the reduced efficiency from adding another switching regulator to the system.
I'm not seeing a good argument for this. You'd be better off with rechargeable batteries.
If one has a device which will draw 20mA continuously at any voltage above the minimum required for operation, and will work equally well at any such voltage, a buck-boost switcher that scan scale a battery's voltage up or down so the device always sees that minimum voltage may both reduce the amount of current drawn from batteries which output more voltage than the device would need, and allow continued operation with batteries which produce less voltage. A win-win.
A buck-boost switcher which raises the voltage significantly above what the device would need for operation will waste energy whenever the battery voltage is between what the device needs and what the booster gives the device.
If the useful performance of the device varies with voltage, scaling the battery voltage up may offer enhanced performance at the cost of reduced battery life; scaling it down may offer better battery life in exchange for reduced performance.
If the device draws power intermittently, and the amount of time it requires power will vary with voltage (e.g. it's a motor which periodically needs move something a certain distance) the amount by which scaling the voltage increases or decreases the current drawn from the battery may be larger or smaller than the amount by which it affects the duration.
If the device has a switching supply built into it, adding a second one in front of it may offer little benefit.
In short, there will be some cases where adding a switching supply may greatly improve battery service life; there will be others where it is useless or counterproductive.