At what temperature does a WiFi thermostat stop sending a signal to the router?
You're looking for a specification called the operating temperature range of the thermostat or router or both. This is the temperature range within which the manufacturer states that the device will function correctly.

For consumer electronics like these, it's typically from 0&deg;C (32&deg;F) up to about 60&deg;C (140&deg;F) to 70&deg;C (158&deg;F). So if you turn the heat to off, the ambient temperature in the house will drop below that range, and the thermostat will stop working properly.

There should be a specifications page in the manuals for the thermostat and router that will tell you their minimum operating temperatures. Keep your house above the upper of those temperatures and you should be OK.