Inhomogeneous strain may develop in hybrid organic metal-halide perovskite thin films due to thermal expansion mismatch with a fabrication substrate, polycrystallinity or even light soaking. Measuring these spatially varying strains is difficult but of prime importance for understanding the effects on carrier mobility, non-radiative recombination, degradation and other optoelectronic properties. Local strain can be mapped using the shifts in vibrational frequencies using Raman or infrared microscopy. We use density functional theory to investigate the effect of uniaxial strain on the vibrations of pseudo-cubic methylammonium lead iodide (CH$_3$NH$_3$PbI$_3$), and identify the vibrational modes most favorable for local strain mapping (86 cm$^{-1}$, 97 cm$^{-1}$, 1457 cm$^{-1}$, and 1537 cm$^{-1}$) and provide calibration curves. We explain the origin of the frequency changes with strain using dynamical matrix and mode eigenvector analysis and study strain-induced structural changes. We also calculate mode Gr\"uneisen parameters, giving information about anharmonicity and anisotropic negative thermal expansion as recently reported for other phases. Our results provide a basis for strain mapping in hybrid perovskites to further the understanding and control of strain, and improve stability and photovoltaic performance.