There are many different "ESP8266" related products. In fact, when you say "ESP8266" you could be talking about the IC -- but, since you also used the term "board", that at least alludes to one of the modules utilizing the ESP8266 IC.
So, which module are you talking about?
I've only, so far, worked with the ESP-01. There is no "Vin" on that module, so I'm unfamiliar. I did a bit of Googling, but, because there are so many different versions, I wasn't able to filter out much info on an ESP module Vin's characteristics, other than: I only saw 5V being applied.
But, you want to know about efficiency. The most efficient way to go is to use a DC-DC converter to transform the battery voltage to the required 3.3V (and then apply that to the VCC or "3.3V" pin). That way, even as the battery voltage sags, the final voltage will stay at 3.3[different chemistries have different discharge curves -- some flatter than others -- plus the discharge rate plays a role]. A converter will mitigate voltage changes as the battery discharges AND make the most efficient use of the stored energy.
BUT
Because use will be intermittent, perhaps this level of efficiency isn't as crucial. A cheaper/easier solution may be to use a larger battery. But, you will, likely, still need regulation, so at least use a linear regulator, and if the battery is only a few tenths of a volt away from 3.3V, as would be the case with a lithium chemistry battery, then use a Low Dropout [LDO] regulator. OR, you can probably get away with a lithium type battery with no regulator -- but check the datasheet for the ESP device you will be using, to see what the max VCC is -- and make sure the battery you choose can never have a higher voltage than that. Also, consider the lowest battery voltage, too.
BUT
ESP modules, being WiFi devices, draw a LOT of current. So, even with intermittent use, there still may be a significant power loss in a linear regulator -- so, were back to the DC-DC converter [i.e. "Switch Mode" solution]. AND, because ESP modules draw a lot of current, make sure that the battery you choose can deliver that much current. Another factor in the efficiency equation, is the batteries internal resistance. The batteries internal resistance will consume power, so you want that resistance to be low enough to not have a significant impact -- over the entire discharge curve. In other words, as a battery drains, the internal resistance increases, so take that into account, as well. OR because of the intermittent nature of this application, a super capacitor [SuperCap] might be of value -- you could get away with a battery that has a higher internal resistance by putting a supercap across it. The battery will charge the SuperCap, then when current demand is high, the SuperCap will deliver it with practically no internal resistance! The result is greater efficiency with a smaller battery. BUT, there will be power loss everytime the battery charges the SuperCap, so to make that more efficient, insert a DC-to-DC converter [probaby a boost converter, but depends on the relative voltages]. BUT, you will need, either a SuperCap with a voltage rating greater than the highest voltage in question [and a little higher as a safety margin] or you will need several SuperCaps in series, and that gets into charge balancing and stuff -- i.e. more complexity -- is it worth the extra efficiency and/or cheaper battery? Only a study will tell. But, there are 5V SuperCaps [which, BTW, can be used in a 5V application -- i.e. no need for that "safety margin" I spoke of before -- whew! Time to bust out the pruning shears!], and they aren't that expensive, so worth a try? [there are other caveats and complexities, but the BUTs are getting out of hand! -- WHERE ARE THOSE PRUNING SHEARS?!?]
BUT
You are, currently, holding most of the cards -- so you will either need to provide us more information [such as the type of ESP module you are currently clutching], or you will need to skurry off and design this thing yourself, using the information I've provided so far. ;D
