Genomic epidemiology is the use of pathogen genomic data to study the spread of a disease in a population. In this study, we investigated the spread of the SARS-CoV-2 virus in Iraq and the effectiveness of targeted travel restrictions implemented by the Iraqi government. Using Bayesian phylogenetic methods, we reconstructed a time-resolved phylogeny and the geographic spread of the virus between Iraq and other regions in the world. We found that the majority of the introductions into Iraq originated from Southern Asia and the Middle East. We also found that the first introductions of the Delta variant into Iraq occurred months before travel restrictions were implemented and that these introductions resulted in ongoing community transmission. We show that new virus introductions were not effectively halted during the period with travel restrictions, with the majority of introductions occurring from Western and Southern Asia. Our findings suggest that travel restrictions targeting a single source country are unlikely to work for a rapidly spreading pathogen in a highly connected world with limited genomic surveillance.