You have two tasks you need to do here:
A) you need to identify the IR signals that the remote sends, and then
B) you need another remote that can send those signals based on the temperature.
There are remote controls that can do #A (learn new codes) but I don't know of anything off the shelf that will do both those things.
,,,
First of all--it is entirely possible that there may be a remote control for these kinds of heaters out there specifically to do this. You might try shopping around on some china-direct shopping sites like aliexpress for an hour or two and see what you can find.
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If you wanted an electronics project, then you could do this on an arduino (or similar device) in two steps:
1) first you set up the arduino with an IR sensor as a "recorder", that will detect the remote control IR blink patterns and report them over the serial connection to the serial monitor on the computer. You use this to discover what codes the remote control is sending, and just write them down at this point. It would be a series of IR [on] states, with short or long time delays in-between.
2) then you re-wire the same arduino with an IR emitter and a temperature sensor, and some way of setting a target temperature. And you write a sketch to turn the heater on and off according to the temperature, using the remote-control codes you discovered in #1.
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Another possible option--one thing I know for certain is available is thermal fan controllers, which can be had for as little as $10 US. These usually have a SPDT relay that will turn another device [off] or [on] when a set temperature is reached, so you can wire them to control a heater or a cooler/fan.... The problem here is that you may need to bypass the normal heater controls, and the heater is gas-fired, which complicates things. With gas heaters there is a feedback-enabled lighting cycle and a thermocouple valve involved to ensure that the gas doesn't keep running if the heater isn't lit. You would need to make certain that it still worked correctly after modifying it, before leaving the thing unattended.
I wouldn't think twice about modifying an electric heater this way, but cutting into a gas heater control setup is a little bit scary in my opinion. I would feel a lot safer doing the arduino-remote-controller thing here because it doesn't interfere with the gas heater's built-in controls.