When food is scarce, the body slows its metabolic rate to conserve energy. Are there any other systems or processes that prioritize which organs receive nutrients?
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:
- Inadequate tear production
- Enlarged/tender liver
- Muscle atrophy
- Rashes, sores, or peeling skin
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.
Glucose is prioritized for the brain and erythrocytes over the muscle and adipose tissue, for example, by hormonal control.
The hormones insulin and glucagon respond to strarvation, insulin secretion falling and that of glucagon increasing. The glucose transporter, GLUT4, in muscle and adipose tissue is dependent on insulin, so that in starvation less-essential uptake of glucose to these tissues declines. The glucose transporters of erythrocytes and brain (e.g. GLUT1) do not depend on insulin, so the supply of glucose to these tissues, which absolutely require it, is not cut off.
Hormones and Starvation: http://www.ncbi.nlm.nih.gov/books/NBK22414/
Insulin and the GLUT4 Glucose Transporter: https://en.wikipedia.org/wiki/GLUT4
GLUT1, insulin-independent Glucose Transporter: https://en.wikipedia.org/wiki/GLUT1
The brain is the most metabolically active organ and will sacrifice the body for fuel.
starvation is characterised by the catabolism of muscle tissue for gluconeogenesis, and the provision of glucose and amino acids for fuel.
Before this occurs, the liver, when glycogen depleted, will switch from creating glucose, to creating ketone bodies from fatty acids. Ketosis is not starvation.
Ketones are eventually preserved for the brain, heart and lungs, whereas the rest of the body can utilise fatty acids, while certain glucose-dependent organs such as the kidneys will still utilise glucose.
Starvation will only occur when there are no free fatty acids to convert to ketone bodies, and the sacrifice of muscle tissue will begin.