Wound dressings able to deliver topically bioactive molecules represent a new generation of wound-regeneration therapies. In this article, foams based on methylcellulose cross-linked with Manuka honey were used as a platform to deliver borate bioactive glass particles doped additionally with copper. Borate bioactive glasses are of great interest in wound-healing applications due to a combination of favorable features, such as angiogenic and antibacterial properties. The multifunctional composite providing the dual effect of the bioactive glass and Manuka honey was produced by freeze-drying, and the resulting foams exhibit suitable morphology characterized by high porosity. Moreover, the performed tests showed improved wettability and mechanical performance with the addition of bioactive glass particles. Dissolution studies using simulated body fluid and cell biology tests using relevant skin cells further proved the excellent bioactivity and positive effects of the foams on cell proliferation and migration. Most interestingly, by the dual release of Manuka honey and ions from the copper-doped bioactive glass, an antibacterial effect against E. coli and S. aureus was achieved. Therefore, the multifunctional foams showed promising outcomes as potential wound dressings for the treatment of infected wounds.