Affordable Housing Preservation Act of 2021 This bill directs the Department of Housing and Urban Development (HUD) to implement a demonstration program for the establishment or preservation of limited- or zero-equity resident-controlled housing. Limited- or zero-equity resident-controlled housing (e.g., a housing cooperative) restricts the initial and resale price of shares of stock or membership interests in a residence so that the shares or interests remain affordable to families of certain incomes. Specifically, HUD may assist nonprofit organizations to establish or preserve such housing through providing (1) mortgage insurance; and (2) grants for predevelopment costs, monitoring fees, technical assistance, gap funding, and intermediary financing. Housing receiving such assistance must be governed by a board of directors elected by members of a limited- or zero-equity resident-controlled housing corporation with voting structured equitably among all members. A limited- or zero-equity resident-controlled housing corporation may sell corporate shares in such housing to an eligible household for a purchase price that ensures dwelling units in the housing will remain affordable for at least 30 years to households earning less than a certain income.