Zika virus (ZIKV) is a mosquito-borne virus associated with neurological disorders such as Guillain-Barré syndrome and microcephaly. In humans, ZIKV is able to replicate in cell types from different tissues including placental cells, neurons, and microglia. This intricate virus-cell interaction is accompanied by virally induced changes in the infected cell aimed to promote viral replication as well as cellular responses aimed to counteract or tolerate the virus. Early in the infection, the 11-kb positive-sense RNA genome recruit ribosomes in the cytoplasm and the complex is translocated to the endoplasmic reticulum (ER) for viral protein synthesis. In this process, ZIKV replication is known to induce cellular stress, which triggers both the expression of innate immune genes and the phosphorylation of eukaryotic translation initiation factor 2 (eIF2α), shutting-off host protein synthesis. Remodeling of the ER during ZIKV replication also triggers the unfolded protein response (UPR), which induces changes in the cellular transcriptional landscapes aimed to tolerate infection or trigger apoptosis. Alternatively, ZIKV replication induces changes in the adenosine methylation patterns of specific host mRNAs, which have different consequences in viral replication and cellular fate. In addition, the ZIKV RNA genome undergoes adenosine methylation by the host machinery, which results in the inhibition of viral replication. However, despite these relevant findings, the full scope of these processes to the outcome of infection remains poorly elucidated. This review summarizes relevant aspects of the complex crosstalk between RNA metabolism and cellular stress responses against ZIKV and discusses their possible impact on viral pathogenesis.