How do I measure the analogue in voltage of a ICL7136 using an 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.