This link on starvation describes the various hormones that start up and shut down during various stages of starvation. This is at a macro level, and involves organs like the liver and pancreas. If your cyborgs include these organs, then the results will be similar.
A cascade failure will occur that results in fat being converted to glucose. When that runs out, protein (muscle) will be converted to glucose. Along the way, your cyborg's organs will begin to fail.
If your system relies on artificial organs, they won't fail. If these organs are incapable of responding to the various cellular signals, then the process may not result in muscle conversion to glucose. But that would just accelerate the death of your beings' living tissues, since cells require glucose to survive.
Your cyborgs can survive without eating in the typical mammalian sense. But the living tissues must take in water, nutrients, oxygen, and glucose in a properly maintained balance. In living beings, a complex series of systems do all of this.
If you take those systems out of your cyborgs, then you will need to replace it with some sort of equivalent mechanical system. This reminds me of the scene in the original RoboCop where Robocop is eating baby food. His metal system had a battery. His living systems still needed energy to fuel those living cells. You could have some mechanical organ that combines compressed oxygen with a pre-mixed solution of glucose, water, and nutrients, and then circulates that mix to your tissues.
But the tissues must have these things or they die. Cells don't survive without them. See also cellular metabolism.
Also, cells don't really live all that long on their own, even with proper nutrients. They are constantly dividing and dying, replacing old cells with new. So the tissues your cyborg leaves the factory with will be constantly replaced throughout it's design life with new tissues. This is called Apoptosis.