Federal Disaster Housing Stability Act of 2020 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. 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, or (3) otherwise require the tenant to vacate the dwelling. These restrictions shall be in place until the declared emergency or disaster ends or the unemployment rate for the affected state remains below a certain threshold for two consecutive months, whichever occurs later.