Identifying host factors is key to understanding RNA virus pathogenicity. Besides proteins, RNAs can interact with virus genomes to impact replication. Here, we use proximity ligation sequencing to identify virus-host RNA interactions for four strains of Zika virus (ZIKV) and one strain of dengue virus (DENV-1) in human cells. We find hundreds of coding and non-coding RNAs that bind to DENV and ZIKV viruses. Host RNAs tend to bind to single-stranded regions along the virus genomes according to hybridization energetics. Compared to SARS-CoV-2 interactors, ZIKV-interacting host RNAs tend to be downregulated upon virus infection. Knockdown of several short non-coding RNAs, including miR19a-3p, and 7SK RNA results in a decrease in viral replication, suggesting that they act as virus-permissive factors. In addition, the 3'UTR of DYNLT1 mRNA acts as a virus-restrictive factor by binding to the conserved dumbbell region on DENV and ZIKV 3'UTR to decrease virus replication. We also identify a conserved set of host RNAs that interacts with DENV, ZIKV, and SARS-CoV-2, suggesting that these RNAs are broadly important for RNA virus infection. This study demonstrates that host RNAs can impact virus replication in permissive and restrictive ways, expanding our understanding of host factors and RNA-based gene regulation during viral pathogenesis.