I have been having issues in my reproduction of Dave's DC load as well, but mine are of the flavour that the pass-mosfet bangs on or off with a very very small window of adjustment (2.5V into the feedback op-amp turns it on, and the mosfet saturates at 3-4V into the feedback op-amp). I thought that this might be a function of the type of mosfet I am using, and specifically the VGS threshold (I was initially using a logic-level mosfet). I swapped this out for a higher VGS mosfet and gt the same thing. I am using a LM358 in the same basic voltage follower -> feedback/control op-amp configuration as Dave.
But to answer the question, the pass 'fet/transistor are there to absorb the power (voltage drop X current) that the sense resistor is not.
Example: if you have a 1Ohm sense, and feed 10V into the load and want to set the load to 1A, the sense resistor should be dropping 1V across it (Ohms Law), and there for the remaining power of the 9V drop @1A must be dissipated somewhere, and this is in the pass 'fet/transistor.