Tested to make sure that even when only the AD586 was heated that there was no change in the reading. I did so by placing the AD586 away from the rest of the circuit and blowing hot air only on it. There was, in fact, no change.
I didn't realize that the LT1012 needed an additional input to function. I added in +15V and -15V to pins 7 and 4, respectively and placed a 10k resistor and the numbers make more sense now. The voltage showed a negative value, but it became more negative as I heated it up (it's negative due to how a transimpedance amplifier works).
I have attached the new setup and the value it shows (~-3V). It became more negative by about 0.4V, which translates to about a 40K increase in temperature, which makes sense.
In addition, I have attached an updated sketch of the circuit. I will need to do more testing to determine how accurate this is. The next step would be to calculate how much noise there is currently and figure out how to remove this circuit from the breadboard and use a PCB or something like that for final testing in an insulated container.
The reason I chose AD743 initially for the OP amp is because at low frequencies (which is what we are working with), a FET amp such as AD743 will have a low current noise at high impedance, which is what we have in this case. While a FET amp has high voltage noise compared to other OP amps, the current noise becomes more important at high impedance, so it will work better. According to Zach's graphs, the AD743 is best at high impedances, followed by LT1012.
Decided to try adding in an OP amp just to see if it would work. Added LT1012 and a 100k resistor to the circuit (I originally wanted to do AD743 as it seems to be the best choice according to Zach's elog here, but it said that they are very precious so I went with LT1012 for testing purposes). When heating it with a heating gun, the output voltage went down by a few 0.01V. The maximum voltage was 0.686V. Similar thing happened when I switched to a 10k resistor, where the maximum was 0.705V and it also went down by a few 0.01V upon heating.
I've attached a few pictures showing the circuit.