We installed a new boiler last year and the board kept throwing error codes to the point where it would shut down.
We couldn't leave the house for more than 24 hours for fear the boiler would lock up in the middle of winter.
Coincidentally we had sprung for a Honeywell WiFi thermostat and I could manually check the setting remotely on my phone very easily.
The thermostat would also send an alert through the app if the temp exceeded parameters. This, coupled with a new DropCam aimed at the boiler's control panel (which did NOT have the ability to alert us other than throwing a code and flashing a red light) allowed us to leave for trips and contact the installer for repairs if needed and we had a neighbor let them in. Never ended up needing that as they eventually fixed the boiler.
The WiFi Thermostat has come in handy in other ways since then:
We installed one in the apartment we rent out to replace the crappy one that the tenant could never figure out how to use. Now when they call or text about issues with Heating or Cooling I can instantly check their thermostat remotely and "fix" it.
We also installed one in my MIL's home. She lives alone and keeps screwing with the thermostat. She is supposed to leave the FAN ON all the time so the humidifier runs even when the forced air heat is not supposed to come on. Instead of driving over there to check it, *again* I just check it via the app on my phone. I can turn the FAN on/off, check settings, re-do them if necessary.
No issues with reliability after more than a year of operation.