Controlling millivolt fireplace with Wi-Fi wall switch
AC (coil) Relays are a thing. Thats the easiest way to do it. There are AC SSRs too, but no cheaper.

You can also do it with optocouplers. This kind (LTV8141) has 2 leds so it runs off AC. Of course the output will momentarily go off at 100Hz, which probably doesn't matter, but a capacitor might be needed on the output







simulate this circuit &ndash; Schematic created using CircuitLab