Can't get wifi thermostat to work with gas boiler
You'll need a relay here

Your boiler, in addition to using nonstandard wire nomenclature, does not provide 24VAC power directly to the thermostat.  Instead, power to the thermostat circuit is supplied through the parallel combination of a control relay contact and the NC contact on the combustion pressure switch.  This means that when the thermostat stops calling for heat, 24VAC power to the 'stat is cut off until the pressure switch opens again.  This is fine for a mechanical thermostat like your current one, but a smart thermostat like the Sensi will get its brain turned to mush every time the boiler kills its power supply.

As a result of this infelicity of the HE Series III's control wiring scheme, you will need to fit a HVAC relay (often sold as a "fan relay" -- these will have a 24VAC coil and at least one NO contact on them) to your system to isolate the thermostat from the boiler's thermostat circuit.  The relay needs to be mounted in a suitable place in the boiler's wiring compartment, and is wired as follows, although you may need to crimp 1/4" female quick-connect/tab terminals onto the wires going to the relay to make the connections to it, depending on the relay you get of course:


One relay coil terminal to the white wire from the thermostat cable
The other relay coil terminal connected to a jumper nutted with the black wire from the thermostat cable and another jumper that goes off to the C terminal on your boiler
One relay contact terminal (the COM terminal) jumpered to the Y terminal on your boiler
The other relay contact terminal (the NO terminal) jumpered to the G terminal on your boiler
And the red wire from the thermostat cable connected through to the R terminal on your boiler


Once this is done, you can then wire the Sensi as follows:


White thermostat wire goes to W (to drive the relay coil with 24VAC on a call for heat)
Red thermostat wire goes to R (to power the thermostat, using the boiler's transformer)
Black thermostat wire goes to C (to provide the 24VAC return for thermostat power)


This way, the Sensi will operate the new relay on a call for heat, closing the thermostat circuit as a result, while having constant 24VAC power available to it, so it doesn't lose its brain every time it stops calling for heat.