Affordable and Homeless Housing Incentives Act of 2023 This bill allows nonrecognition of gain, for income tax purposes, from the sale of real property to a qualified housing operator for use or development by such operator as affordable housing. The bill defines qualified housing operator to include a state or local government, a tribally designated housing entity, a community housing development organization, or a tax-exempt entity organized to provide affordable housing and which receives federal, state, or local grants to develop or operate such housing. To be eligible for nonrecognition of gain, the seller must make the property either (1) residential rental property, not less than 75% of the units of which are affordable to low-income families; or (2) a homeless shelter. The Departmental Enforcement Center of the Department of Housing and Urban Development must ensure that sales of real property under this bill meet certain involuntary conversion requirements.