The Due is the only Arduino board with a built in DAC. (Two of them.) If you need more than two outputs or want to use a board such as the Uno, then you need to consider building a DAC.
Option 1: PWM
Using a RC-filter you can create a low-pass filter that allows you to create true analog voltages from a PWM pin. This method could be difficult to calibrate or may not work depending on the impedance of the circuit you are driving.
Option 3: DAC Chip
The most efficient, but most expensive, method would be to use an external DAC chip. You can pick one that is designed specifically for the voltages you need and potentially find one with multiple outputs. Regardless, most DAC chips feature a SPI or I2C interface, so you can chain multiple chips to a single Arduino.
It looks to me like the chip has reference voltage pins and then sensor voltage pins. So it is reading voltage changes from a sensor that reports back in that way, massaging those levels (amplification, scaling, whatever), and then outputting to the display.
Yes, I would think logically reading the same sensor outputs that the chip is reading on IN HI and IN LOW would allow you to access the data. I am assuming the mere act of connecting to those pins and reading them won't change the voltage, but I would defer on that to those more experienced as I could be wrong.
I think your issue, however, is going to be doing the scaling and amplification and conversion of the signals coming in so that the Arduino and the chip both think a given voltage represents the same (in your case) temperature reading. Once they both say that particular voltage level is that many degrees then it becomes a data logging issue.