Federal Disaster Housing Stability Act of 2021 This bill imposes limits on foreclosures and evictions in areas affected by a declared disaster or emergency. A servicer of certain federally backed residential mortgages for a property in such an area may not initiate the foreclosure process or take other foreclosure-related actions relating to that property unless the property is vacant or abandoned. These restrictions shall be in place for six months after the disaster or emergency is declared. Similarly, the landlord of a dwelling in such an area may not (1) initiate eviction proceedings against a tenant in that dwelling for nonpayment of rent or fees; (2) charge that tenant a penalty for nonpayment of rent; (3) increase the rent; (4) prevent a tenant who has temporarily relocated from returning; or (5) otherwise require the tenant to vacate the dwelling. Landlords must also provide notice to tenants regarding available federal assistance. These restrictions shall be in place for 90 days beginning (1) when the disaster or emergency is declared; or (2) in the case of a dwelling eligible for assistance, when such assistance is received or when eligibility for such assistance has expired.