Adding an extra thing to the device means we have to make sure there's enough power available to run it.
The power hungry ESP8266 can draw up to 220mA or so in 802.11b mode, CCK 1Mbps at max power. In my instance, since the EnviR is about 5cm away from my AP, we could certainly reduce the power by using a less power hungry mode. But since these ESP8266 chips seem to have an amazing range it would seem a shame not to be able to use it !
The power coming from the plugtop supply that comes with the EnviR is 3V unregulated rated to 80mA, and on the EnviR's main board we can see a little PSU section with a SOT-23 regulator (what's that, HFEK? looks like a linear regulator, probably a LDO 3.3V of some sort?) and a couple of decoupling caps for good measure (photo purloined from Jack Kelly's teardown here).
The PIC18F86J90 (see a photo of it here, from Jack Kelly (not me)) which runs the main board can run on anything from 2.0V to 3.6V, and the ESP8266 can run between 1.275V and 3.6V. So whatever voltage that regulator is outputting should be fine for the ESP.
If I replace the plugtop PSU with something a bit beefier that can deliver, say, 300mA, (like this one from Rapid, perhaps - 3.3v 1.5a Mini Plugtop Sw Mode Psu ) then the question then is whether that PSU section on the main board can provide enough current or not. Chances are the regulator is capable of at least a couple hundred mAs, but we'll have to test it. If it can't provide enough current , we'll have to add our own regulator circuit on the I/O board - something I'd like to avoid if at all possible.
Further measurement and poking is required.