When food is scarce, the body slows its metabolic rate to conserve energy. Are there any other systems or processes that encourage prioritization of organs?
The glycogen in the liver begins providing blood glucose. Muscle glycogen is used as fuel by the muscles, fat cells (adipose tissue) release fatty acids to manufacture ketone bodies in the liver and to be used by the brain as fuel, and body proteins are converted to glucose.
In short, the body's metabolism shifts to catabolic reactions.
If this continues for too long, you begin to see effects of starvation:
In short, deficiencies of energy, protein, iron, and zinc can really mess you up in the long term.
The switch from glucose to ketone bodies as the principal blood metabolite accompanies starvation. This prioritizes the heart, which preferentially uses ketone bodies as a fuel (update; actually it's fatty acids, but they're metabolically similar.) This de-prioritizes the brain, which preferentially uses glucose.
 I was asked for a source. This is definitely in Voet, Voet & Pratt's Fundamentals of Biochemistry, but I vouch for Wikipedia's description.