Housing Promotes Livelihood and Ultimate Success Act of 2023 or the Housing PLUS Act of 2023 This bill modifies the requirements under which the Department of Housing and Urban Development (HUD) must award grants under the Continuum of Care Program, which provides funding for nonprofit providers and state and local governments to quickly rehouse homeless individuals and families. Specifically, the bill bars HUD from prohibiting, limiting, or restricting Continuum of Care grants to recipients that (1) require program participants to receive supportive services, (2) require program participants to be sober or drug-free, or (3) are faith-based organizations. The bill also requires that at least 30% of Continuum of Care grant funds be set aside for eligible grant recipients that provide or offer access to wraparound services.