As emerging contaminants, antiviral drugs (ATVs) in water pose an increasing threat to human health and the ecosystem. Using specific and selective adsorption methods to remove ATVs remains challenging due to their trace concentration in real matrices. In this study, pyridine ionic liquid ([BPy][Sal])-modified metal–organic framework (MOF), a type of novel material, was synthesized as [BPy][Sal]@MOF to adsorb ATVs with different molecular structures, i.e., Arbidol (Arb) and Favipiravir (Fav). Arb adsorption followed the pseudo-second-order model, indicating that chemisorption is the dominant process. The Langmuir isothermal model showed that the maximum adsorption capacities toward Arb and Fav were 506.9 and 35.3 mg g−1, respectively, suggesting specific adsorption of Arb on [BPy][Sal]@MOF. Density functional theory (DFT) was used to investigate the specific adsorption mechanism. The molecular electrostatic potential and the intermolecular weak interaction analysis using the DFT method illustrated that the amino hydrogens in MOF could form hydrogen bonds (NHO) with the hydroxyl oxygens in Arb. Simultaneously, the indole ring in Arb could cause π–π interaction with the pyridine ring in [BPy][Sal] part from [BPy][Sal]@MOF. In contrast, the adsorption of Fav only involved π–π interaction between the benzene rings in Fav and the MOF part from [BPy][Sal]@MOF. The adsorption energy and energy gap calculation results verified that the adsorption strength of Arb was higher than that of Fav, and the key role of [BPy][Sal] in [BPy][Sal]@MOF helped achieve specific adsorption was confirmed. [BPy][Sal]@MOF exhibited exceptional recyclability after five adsorption–desorption cycles in environmental waters, demonstrating its excellent adsorption performance in practical applications.