Does the Catholic Church consider not administering a feeding tube passive euthanasia? If a person was near death due to a severe head injury, but now only completely incapacitated (as in, unable to respond to stimuli; eyes open, but not tracking; not able to speak or swallow), is it OK to withhold giving that person a feeding tube?  Does it matter what the odds for improved health are? And, if it is not OK, how does anyone benefit from the person's continued survival?
Yes, in your example, it would be passive euthanasia.
Food and water are not extraordinary means of life support, so withholding those from someone is as immoral as starving or dehydrating any other person to death. However, there are situations where administering food and water may be difficult and actually hasten death, in which case it would be immoral.
From the 2009 Ethical and Religious Directives for Catholic Health Services p. 31:


A person may forgo extraordinary or disproportionate means of preserving life.  Disproportionate means are those that in the patient’s judgment do not offer a reasonable hope of benefit or entail an excessive burden, or impose excessive expense on the family or the community.

In principle, there is an obligation to provide patients with food and water, including medically assisted nutrition and hydration for those who cannot take food orally. This obligation extends to patients in chronic and presumably irreversible conditions (e.g., the “persistent vegetative state”) who can reasonably be expected to live indefinitely if given such care.40 Medically assisted nutrition and hydration become morally optional when they cannot reasonably be expected to prolong life or when they would be “excessively burdensome for the patient or [would] cause significant physical discomfort, for example resulting from complications in the use of the means employed.”41 For instance, as a patient draws close to inevitable death from an underlying progressive and fatal condition, certain measures to provide nutrition and hydration may become excessively burdensome and therefore not obligatory in light of their very limited ability to prolong life or provide comfort.



For ordinary vs. extraordinary means of life support, see §6.7 of Health Care Ethics by Benedict Ashley, O.P.