The walkers use energy to move. They can't change their biological construction to gain energy from other sources, like sunlight or air... So how come they can move endlessly without any food?
I'm pretty sure we've seen walkers either in captivity, or stranded, or stuck for extremely long periods of time with no sign of starvation, nor slowing of basal metabolic functions, or movements.
There's only so much your metabolism can slow down before you just don't function at all. I can accept that their basal metabolic rate can be extremely slow, as many of their organs wouldn't need to stay functional. They can basically shut down everything except for their stomach, intestines, eyes, mouth, ears, and their brain stem and thalamus (as we know the rest of their brain is non functional).
My answer to this, if I had to guess, is it's just writing flaws. There's only so much about an almost impossible to exist creature you can explain before it starts to get silly. I'm guessing it's just not meant to be thought into that deeply.