(1) Background: Few studies to date have assessed the influences induced by climate change on the spatial distribution and population abundance of Aedes albopictus using the latest climate scenarios. In this study, we updated the current distribution of Ae. albopictus mosquitoes and evaluated the changes in their distribution under future climate conditions, as well as the risk of dengue virus emergence in Romania. (2) Methods: Under the two scenarios: High scenario (HS) when no drastic measures to reduce the effects of global warming will be taken, or they are not effective and low scenario (LS) when very stringent greenhouse control measures will be implemented. (3) Results: The results estimate an increase in temperatures in Romania of up to 2.6 °C in HS and up to 0.4 °C in LS, with an increase in the period of virus replication within the vector from June to October in HS and from May to September in LS. Moreover, in 2022, Ae. albopictus was reported in a new county, where it was not identified at the last monitoring in 2020. (4) Conclusions: The rapid spread of this invasive species and the need to implement monitoring and control programs for the Aedes population in Romania are emphasized.