This is the first of a series of papers on the Infrared Database of Extragalactic Observables from Spitzer (IDEOS). In this work, we describe the identification of optical counterparts of the infrared sources detected in Spitzer Infrared Spectrograph (IRS) observations and the acquisition and validation of redshifts. The IDEOS sample includes all the spectra from the Cornell Atlas of Spitzer/IRS Sources (CASSIS, Lebouteiller et al., 2011ApJS..196....8L, 2015ApJS..218...21L) of galaxies beyond the Local Group. Optical counterparts were identified from correlation of the extraction coordinates with the NASA Extragalactic Database (NED). To confirm the optical association and validate NED redshifts, we measure redshifts with unprecedented accuracy on the IRS spectra ({sigma}({Delta}z/(1+z))~0.0011) by using an improved version of the maximum combined pseudo-likelihood method (MCPL). We perform a multistage verification of redshifts that considers alternate NED redshifts, the MCPL redshift, and visual inspection of the IRS spectrum. The statistics is as follows: the IDEOS sample contains 3361 galaxies at redshift 0<z<6.42 (mean: 0.48, median: 0.14). We confirm the default NED redshift for 2429 sources and identify 124 with incorrect NED redshifts. We obtain IRS-based redshifts for 568 IDEOS sources without optical spectroscopic redshifts, including 228 with no previous redshift measurements. We provide the entire IDEOS redshift catalogue in machine-readable formats. The catalogue condenses our compilation and verification effort, and includes our final evaluation on the most likely redshift for each source, its origin, and reliability estimates.