This condition is called Exertional or Capture Myopathy.
Specifically, according to the Manual of Common Diseases and Parasites of Wildlife in Northern British Columbia
Exertional or capture myopathy (disease of the muscle) is a
non-infectious disease of muscles that is characterized by damage to
muscle tissues brought about by physiological changes, usually
following extreme exertion, struggle and/or stress.
It may be seen peracutely (immediately), acutely (within minutes) or chronically (in hours, days or weeks).
The House Rabbit Society of Georgia describes the mechanics of the disease in rabbits over long term:
The perception of a dangerous or frightening situation causes the
release of chemicals called neurotransmitters from the pituitary gland
at the base of the brain. These neurotransmitters affect many tissues
in the rabbit’s body, but most notably the adrenal glands that release
epinephrin (adrenaline) and, with protracted periods of stress,
glucocorticosteroids.
Epinephrine causes the rabbit’s heart rate and blood pressure to
increase. Blood flow is directed to vital muscles and organs and away
from those that are nonessential in this dangerous situation. The
rabbit’s respiratory rate increases, his eyes dilate, and his blood
sugar (the fuel for the bodies tissues) soars. Other neurotransmitter
and hormonal effects of fear not as easily understood are that they
cause an ileus of the gastrointestinal (GI) tract-that is, they cause
the GI tract to stop moving.
....when these physiological conditions exist for a long period,
they affect the rabbit negatively. Restriction of blood supply to
“nonessential tissues” leads to their dysfunction. Ileus results in
changes in GI tract bacterial balance and potentially can lead to gut
stasis, diarrhea, enteritis or even enterotoxemia. Exhaustion of liver
energy stores leads to a starvation of body tissues that may be
lethal.
For the short term, death seems to be caused by what we intutitively expect - adreneline's effect on the heart. A study conducted on rabbits and rats reports (in the abstract):
Arterial hypertension, hypotension, and heart failure are the
reactions typical of the stress-sensitive animals. The main cause of
sudden death under emotional stress is the abrupt decrease in
peripheral vascular resistance. Adrenal hormones are crucial to the
mechanisms of sudden death under emotional stress.
