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Context. NGC 6441 is a bulge globular cluster (GC) with an unusual horizontal branch morphology and a rich population of RR Lyrae (RRL) and Type II Cepheid (T2C) variables that is unexpected for its relatively high metallicity. Aims. Our goal is to characterize the pulsation properties of the population II RRL and T2C variables in this metal-rich GC and compare them with a sample of variables in more metal-poor clusters, as well as with theoretical predictions. Methods. We present JHK_s time-series observations of 42 RRLs, eight T2Cs, and ten eclipsing binary candidate variables in NGC 6441 . The multi-epoch observations were obtained using the FLAMINGOS-2 instrument on the 8-m Gemini South telescope. Multiband data are used to investigate pulsation properties of the RRL and T2C variables, including their light curves, instability strip, period-amplitude diagrams, and period-luminosity and period-Wesenheit relations (PLRs and PWRs) in the JHK_s filters. Results. The near-infrared (NIR) pulsation properties of RRL variables are well fitted with theoretical models that have canonical helium content and the mean-metallicity of NGC 6441 ([Fe/H] = -0.44 ± 0.07 dex). The helium-enhanced RRL models predict brighter NIR magnitudes and bluer colors than the observations of RRL in the cluster. We find that the empirical slopes of the RRL PLRs and PWRs in NGC 6441 are statistically consistent with those of RRLs in more metal-poor GCs, as well as the theoretical models. Therefore, we use theoretical calibrations of RRL period-luminosity-metallicity (PLZ) relations to simultaneously estimate the mean reddening, E(J - K_s)=0.26 ± 0.06 mag, and the distance, d = 12.67 ± 0.09 kpc, to NGC 6441. Our mean reddening value is consistent with an independent estimate from the bulge reddening map based on red clump stars. The RRL-based distance agrees well with similar literature measurements based on the PLZ relations, and the Gaia and other independent methods. Our distance and reddening values provide a very good agreement between the PLRs of T2Cs in NGC 6441 and those for RRLs and T2Cs in Galactic GCs that span a broad range of metallicity. Conclusions. We conclude that the NIR colour-magnitude diagrams, pulsation properties, and PLRs for RRLs and T2Cs in NGC 6441 are in good agreement with the predictions of RRL pulsation models with canonical helium content. This suggests that these population II variables are either not significantly helium enhanced, as previously thought, or the impact of such enhancement is smaller in NIR than the predictions of the pulsation models.