Does Catholic teaching require a feeding tube?

Not automatically, and this is the teaching most often reported in half.

Directive 58 of the Ethical and Religious Directives begins with an obligation in principle to provide food and water, including by medically assisted means, for people who cannot take food by mouth. The same directive then states the limits: these measures "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."

It then applies that to exactly the situation a family with advanced cancer is in: "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."

The 2020 Vatican letter Samaritanus bonus goes further, saying that when nutrition and hydration no longer benefit the patient because the body cannot absorb or metabolize them, "their administration should be suspended."

Whether that describes your relative is a clinical question. Ask the team whether the nutrition is being absorbed and whether hydration is helping or adding to swelling and secretions, and ask for a Catholic chaplain and an ethics consultation together.