We provide a new method for deriving heavy element abundances taking advantage of the unique suite of fine-structure lines in the mid- to far-infrared (IR) range. Using grids of photo-ionisation models that cover a wide range in O/H and N/O abundances and ionisation parameter, our code HII-CHI-MISTRY-IR (HCM-IR) provides model-based abundances based on extinction-free and temperature-insensitive tracers. These are two significant advantages over optical diagnostics when they are applied to dust-obscured regions or unresolved (stratified) nebulae, typical conditions found in high-z galaxies. The performance of the code is probed using three different samples of galaxies that extend over a wide range in metallicity, 7.2 ≲ 12 + log(O/H) ≲ 8.9, with available mid- to far-IR spectroscopic observations from Spitzer and Herschel, respectively. The samples correspond to 28 low-metallicity dwarf galaxies, 19 nearby starbursts, and 9 luminous IR galaxies. The IR model-based metallicities obtained are robust within a scatter of 0.03 dex when the hydrogen recombination lines, which are typically faint transitions in the IR range, are not available. When compared to the optical abundances obtained with the direct method, model-based methods, and strong-line calibrations, HCM-IR estimates show a typical dispersion of ∼ 0.2 dex. This is in line with previous studies comparing IR and optical abundances and does not introduce a noticeable systematic above 12 + log(O/H) ≳ 7.6. This accuracy can be achieved by measuring the sulphur ([S IV]10.5 μm and [S III]18.7, 33.5 μm) and the neon lines ([Ne III]15.6 μm and [Ne II]12.8 μm). Additionally, HCM-IR provides an independent N/O measurement when the oxygen ([O III]52, 88 μm) and nitrogen ([N III]57 μm) transitions are measured. The derived abundances in this case do not rely on particular assumptions on the N/O ratio. Large uncertainties (∼ 0.4 dex) may affect the abundance determinations of galaxies at sub- or over-solar metallicities when a solar-like N/O ratio is adopted. Finally, the code has been applied to 8 galaxies located at 1.8 < z < 7.5 with ground-based detections of far-IR lines redshifted in the submillimetre range, revealing solar-like N/O and O/H abundances in agreement with recent studies. A script to derive chemical abundances with HCM-IR has been made publicly available online.