This paper reports a study of the effect of a bar on the neutral hydrogen (HI) content of starburst and Seyfert galaxies. We also make comparisons with a sample of "normal" galaxies and investigate how well starburst and Seyfert galaxies follow the fundamental scaling Tully-Fisher (TF) relation defined for normal galaxies. 111 Markarian (Mrk) IRAS galaxies were observed with the Nancay radiotelescope, and HI data were obtained for 80 galaxies, of which 64 are new detections. We determined the (20 and 50%) linewidths, the maximum velocity of rotation and total HI flux for each galaxy. These measurements are complemented by data from the literature to form a sample of Mrk IRAS (74% starburst, 23% Seyfert and 3% unknown) galaxies containing 105 unbarred and 113 barred ones. Barred galaxies have lower total and bias-corrected HI masses than unbarred galaxies, and this is true for both Mrk IRAS and normal galaxies. This robust result suggests that bars funnel the HI gas toward the center of the galaxy where it becomes molecular before forming new stars. The Mrk IRAS galaxies have higher bias-corrected HI masses than normal galaxies. They also show significant departures from the TF relation, both in the B and K bands. The most deviant points from the TF relation tend to have a strong far-infrared luminosity and a low oxygen abundance. These results suggest that a fraction of our Mrk IRAS galaxies are still in the process of formation, and that their neutral HI gas, partly of external origin, has not yet reached a stationary state.