Tenant Protection Notification Act of 2020 This bill requires that affected tenants be notified of the 120-day moratorium on evictions that began on March 27, 2020, in response to the COVID-19 (i.e., coronavirus disease 2019) public health emergency. Specifically, the Department of Housing and Urban Development (HUD) must develop a written notice in plain language to inform such tenants that the moratorium applies to their dwelling and to explain its effects. Lessors must provide the notice to the affected tenants. If there is any change to the moratorium during the COVID-19 public health emergency, HUD must develop, and lessors must provide to tenants, an additional written notice to inform tenants of the change and its effects.