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5d transition-metal oxides have been of interest due to their strong spin-orbit interaction, and proposed Dirac crossings in the electronic structure. Such materials, including SrIrO₃, show large changes in electronic structure and properties due to strain. Epitaxial metastable perovskite SrIrO₃ thin films were grown by pulsed laser deposition with in situ high-pressure RHEED on various perovskite oxide substrates producing epitaxial strain from -2 to 1%. We find that the room temperature SrIrO₃ resistivity increases by up to 3x with 3% change in strain. At low temperatures the Hall coefficient changes drastically. This low-temperature Hall signal is increasingly nonlinear with tensile strain, and the Hall resistivity does not scale with thickness. A multiband model reproducing this nonlinearity requires two electron bands, in disagreement with ARPES studies. We therefore propose a magnetic origin and consider the role of octahedral rotations on the electronic transport properties of SrIrO₃ thin films.