Your pic didn't make it through.
Home Automation
Home automation is the residential extension of building automation.
It is automation of the home, housework or household activity.
Home automation may include centralized control of lighting, HVAC (heating, ventilation and air conditioning), appliances, security locks of gates and doors and other systems, to provide improved convenience, comfort, energy efficiency and security.
Warning: Working with electricity can result in injury, property damage, or even death if it is not done properly. Please keep this in mind while assisting others. If you are not sure about what you are doing, hire a licensed professional.
Rules
- No abusive behaviour. This is a forum for friendly discussion; personal attacks will not be tolerated and you will be banned without warning.
- Referral/affiliate links are NOT ALLOWED!
- NO POLITICS! There are plenty of other communities to discuss them; this is not one.
- When posting project details must be included. Posting a video or image without detail will result in a removed post and may result in a ban.
- Crowdfunding links are not allowed.
- Reposts, low-effort content and karma farming may be removed at the discretion of the mods. Posters may be banned without warning.
If your fireplace has a fan then you have power there.
With the model of the fireplace or the module inside, we could tell you in greater detail how it can be switched.
They basically all support a thermostat in addition to a manual switch. Thermostats are frequently in the form of a remote control, with a module that's just plugged in to spade terminals beneath the fireplace. For a dry contact style fireplace the module is battery operated.
Are you trying to mechanically toggle the fireplace switch? That's going to be a though and probably unreliable. Can you mount a low voltage relay (like 12VDC) next to the fireplace switch (as you have pointed out the thermocouple only puts out a few millivolts) and use a WiFi switch powering a 12V DC power adaptor to energize the relay? Just a thought.