I have been measuring the voltage response to current of an LED. It's a point source LED with an oval emission area of 40x150um (link to datasheet https://www.daido.co.jp/en/products/point_source_led/pdf/med7p4_e.pdf). I have done the same measurements for a range of temperatures from 10°C-70°C. I expected the decrease in voltage with temperature to be linear, but these graphs don't show that. Is there a physical explanation for this or is it more likely to be some kind of experimental error?
Those currents are so high that the ohmic resistance of the semiconductor material becomes important. It is not just the voltage over the $pn$ junction. (And another effect is that at high currents, the current will affect the temperature.)
Theoretical values can only be expected in the region where the $IV$ curve is exponential. Maybe up to 1 mA or something like that in such devices.