The Tax oncoprotein encoded by the human T-cell leukemia virus type 1 plays a pivotal role in viral persistence and pathogenesis. Human T-cell leukemia virus type 1-infected cells proliferate faster than normal lymphocytes, expand through mitotic division, and accumulate genomic lesions. Here, we show that Tax associates with the minichromosome maintenance MCM2-7 helicase complex and localizes to origins of replication. Tax modulates the spatiotemporal program of origin activation and fires supplementary origins at the onset of S phase. Thereby, Tax increases the DNA replication rate, accelerates S phase progression, but also generates a replicative stress characterized by the presence of genomic lesions. Mechanistically, Tax favors p300 recruitment and histone hyperacetylation at late replication domains, advancing their replication timing in early S phase.