1 A current source mosfet heats up
low on resistance of .025 ohm (irfz44n).


The Rds(on) in the datasheet applies when the FET is fully switched (aka used in the saturation region). 

This circuit uses the FET in the linear region, so this spec doesn't apply.

Anyway it's very easy to calculate the power consumed by the FET. The current is 1 A. The voltage is 12 V less a tiny amount for the sense resistor. So the power consumption must be very near 12 W. 

With 62 C/W junction-to-ambient thermal resistance (per the IR/Infineon datasheet), that's more than enough let the smoke out of the part.