The reason birds, or any animal cannot eat or drink if injured, is in case they need surgery- which may require a general anaesthetic or sedation to calm the bird to do procedures.
General anaesthesia and many sedatives can cause nausea or even vomiting, even in an unconscious animal. They also relax the muscles around the pharynx, which is where the bird controls whether air goes down the trachea or food and fluid down the esophagus. As the animal is unconscious, it has no control over the swallowing process. This increases the risk of any food or liquid in the birds stomach from coming up the esophagus and being aspirated, or inhaled into the lungs; which is a potentially life threatening condition, or at the least an unwanted complication of aspiration pneumonia.
Fluid is as dangerous as food. In some ways fluid has a great risk of back flowing up the anaesthetized bird's esophagus with the risk of being inhaled.
Upper airway obstruction is common during both anaesthesia and sleep.
Obstruction is caused by loss of muscle tone present in the awake
state. The velopharynx, a particularly narrow segment, is especially
predisposed to obstruction in both states. Patients with a tendency to
upper airway obstruction during sleep are vulnerable during
anaesthesia and sedation. (1)
The upper airway during anaesthesia
Hillman, Platt, Eastwood doi: 10.1093/bja/aeg126 (1)
Avian Digestion PDF
Sturkie's Avian Physiology by P. D. Sturkie, G.Causey Whittow ISBN: 978-0-12-747605-6