Is it permissible to eat and drink from the food and drink provided by an aunt who deposits her money in an interest-based bank and spends on her household from the interest earnings, knowing that refusing to do so will cause estrangement?

If your aunt has another permissible source of income apart from the unlawful money, then there is no harm in eating her food and accepting her hospitality. This is because if lawful and unlawful money are mixed and cannot be distinguished, it is permissible to deal with its owner and eat from it. The Prophet, peace and blessings be upon him, and his companions interacted with the Jews and ate their food, despite knowing that their wealth was not free from unlawful elements.

Some scholars are of the view that money acquired through usury is unlawful only for the one who acquired it, and not for one who received it through a permissible means such as a gift or hospitality.

If refusing the hospitality will affect your aunt and lead her to repent from her unlawful act, then refusing the hospitality is preferable.